13

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Going for a Drive

Learning Goal: To gain a qualitative understanding of kinematics and how the qualitative nature of position and

velocity versus time graphs relates to the equations of kinematics.

In this problem, you will explore kinematics using an applet that simulates a car moving under constant acceleration. When you open the applet, you will see three sliders that allow you to adjust the initial position , the initial velocity , and the acceleration . Set the initial position to 0 , the initial velocity to and the acceleration to 5 . Run the simulation. Notice that, as the movie proceeds, pictures of the car remain at certain points. Once the simulation is over, these pictures form a motion diagram--a representation of motion consisting of pictures taken at equal time intervals during the motion. In this case, the interval between pictures is one second.

Below the movie, the position of the car as a function of time is graphed in green. Run the simulation several times, paying attention to how the graph and the motion diagram/movie of the car's motion relate to each other.

Part A

Which of the following describe the relationship between the motion diagram/movie and the graph?

Check all that apply.

ANSWER: _{When the slope of the graph is close to zero, the pictures in the motion diagram are close together.}

When the slope of the graph is steep, the car is moving quickly.

When the slope of the graph is positive, the car is to the right of its starting position.

When the x position on the graph is negative, the car moves backward. When the x position on the graph is positive, the car moves forward.

When the x position on the graph is negative, the car moves slowly. When the x position on the graph is positive, the car moves quickly.

When the x position on the graph is negative, the car is to the left of its starting position. When the

x position on the graph is positive, the car is to the right of its starting position. All attempts used; correct answer displayed

Notice that the first and second options are always true, regardless of the values of , , and . The last option, however, is only true when . Frequently, you will be able to pick your coordinate system. In such cases, making is often a good choice.

Part B

✔

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Run the simulation, paying close attention to the graph of position. Press reset and change the value of . Run the simulation again, noting any changes in the graph. How does varying affect the graph of position?

ANSWER: _{Increasing increases the width of the graph, whereas decreasing decreases the width.}

Increasing shifts the graph to the right, whereas decreasing it shifts the graph to the left. Increasing shifts the graph to the left, whereas decreasing it shifts the graph to the right. Increasing shifts the graph upward, whereas decreasing it shifts the graph downward. Increasing shifts the graph downward, whereas decreasing it shifts the graph upward. Changing does not affect the graph.

Correct

Part C

Now, run the simulation with different values of , but don't use any positive values. Note any changes in the graph. How does varying affect the graph of position?

Choose the best answer.

ANSWER: _{Increasing increases the width of the graph, whereas decreasing decreases the width.}

Increasing shifts the graph to the right and upward, whereas decreasing it shifts the graph to the left and downward.

Increasing shifts the graph to the left and upward, whereas decreasing it shifts the graph to the right and downward.

Increasing shifts the graph to the right and downward, whereas decreasing it shifts the graph to the left and upward.

Increasing shifts the graph to the left and downward, whereas decreasing it shifts the graph to the right and upward.

Changing does not affect the graph. Correct

This behavior may be a bit difficult to understand by just looking at the equation for position vs. time that you know from kinematics. If you complete the square to get the equation into standard form for a parabola, this should become more apparent.

Now that you've seen how affects the graph, run the simulation with a few different values of acceleration. You should see that increasing the acceleration decreases the width of the graph and decreasing the acceleration increases the width. (Decreasing the acceleration below 0 makes the parabola open downward instead of upward.)

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Enter the equation for position as a function of time. Before submitting your answer, check that it is consistent with the qualities of the graph that you have identified. For instance, if you increase in the equation, would it move the graph upward?

Express your answer in terms of time , initial position , initial velocity , and acceleration . ANSWER:

=

Correct

Part E

Now, open this applet. This applet looks like the previous applet, but when you run the simulation, you will now get graphs of both position and velocity. Run the simulation several times with different values of . How does changing affect the graph of velocity?

ANSWER: _{Increasing increases the slope of the graph, whereas decreasing decreases the slope.}

Increasing shifts the graph to the right, whereas decreasing it shifts the graph to the left. Increasing shifts the graph to the left, whereas decreasing it shifts the graph to the right. Increasing shifts the graph upward, whereas decreasing it shifts the graph downward. Increasing shifts the graph downward, whereas decreasing it shifts the graph upward. Changing does not affect the graph.

All attempts used; correct answer displayed

Part F

Run the simulation again, with the following settings: , , and . The units of time in the graph are seconds. At what time is the velocity equal to zero?

Express your answer in seconds to the nearest integer.

ANSWER: ₃

Correct

Notice that the position graph has a minimum when velocity equals zero. This should make sense to you. Since velocity is the derivative of position, position has a local minimum or maximum when velocity is zero.

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Suppose that a car starts from rest at position and accelerates with a constant acceleration of 4.15 . At what time is the velocity of the car 19.2 ? Use the applet to be certain that your answer is reasonable.

Hint G.1 Choose the kinematic equation

Hint not displayed

Hint G.2 Using the applet to check your answer

Hint not displayed Express your answer in seconds to three significant figures.

ANSWER:

= 4.63 Correct

Part H

For the same initial conditions as in the last part, what is the car's position at time 4.05 ? Again, be sure to use the applet to check that your answer is reasonable.

Hint H.1 Choose the kinematic equation

Hint not displayed Express your answer in meters to three significant figures.

ANSWER:

= 30.7 Correct

Any time that you are working a physics problem, you should check that your answer is reasonable. Even when you don't have an applet with which to check, you have a wealth of personal experience. For example, if you obtain an answer such as "the distance from New York to Los Angeles is 3.96 ," you know it must be wrong. You should always try to relate situations from physics class to real-life situations.

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As you look out of your dorm window, a flower pot suddenly falls past. The pot is visible for a time , and the vertical length of your window is . Take down to be the positive direction, so that downward velocities are positive and the acceleration due to gravity is the positive quantity .

Assume that the flower pot was dropped by someone on the floor above you (rather than thrown downward). Part A

From what height above the bottom of your window was the flower pot dropped? Hint A.1 How to approach the problem

Hint A.2 Find the velocity at the bottom of the window

Hint A.3 The needed kinematic equation

Hint not displayed Express your answer in terms of , , and .

ANSWER:

=

Correct

Part B

If the bottom of your window is a height above the ground, what is the velocity of the pot as it hits the ground? You may introduce the new variable , the speed at the bottom of the window, defined by

.

Hint B.1 Needed kinematic equation

Hint B.2 Find the initial height of the pot

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ANSWER:

=

Correct

A Man Running to Catch a Bus

A man is running at speed (much less than the speed of light) to catch a bus already at a stop. At , when he is a distance from the door to the bus, the bus starts moving with the positive acceleration .

Use a coordinate system with at the door of the stopped bus.

Part A

What is , the position of the man as a function of time?

Hint A.1 Which equation should you use for the man's speed?

Hint not displayed Answer symbolically in terms of the variables , , and .

ANSWER:

=

Correct

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What is , the position of the bus as a function of time?

Hint B.1 Which equation should you use for the bus's acceleration?

Hint not displayed Answer symbolically in terms of and .

ANSWER:

=

Correct

Part C

What condition is necessary for the man to catch the bus? Assume he catches it at time . Hint C.1 How to approach this problem

ANSWER: Correct Part D

Inserting the formulas you found for and into the condition , you obtain the following:

, or .

Intuitively, the man will not catch the bus unless he is running fast enough. In mathematical terms, there is a constraint on the man's speed so that the equation above gives a solution for that is a real positive number.

Find , the minimum value of for which the man will catch the bus. Hint D.1 Consider the discriminant

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Hint D.2 What is the constraint?

Hint not displayed Express the minimum value for the man's speed in terms of and .

ANSWER:

=

Correct

Part E

Assume that the man misses getting aboard when he first meets up with the bus. Does he get a second chance if he continues to run at the constant speed ?

Hint E.1 What is the general quadratic equation?

Hint not displayed ANSWER: _{No; there is no chance he is going to get aboard.}

Yes; he will get a second chance Correct

Ballistocardiograph Conceptual Question

In a ballistocardiograph, a patient lies on an extremely low friction horizontal table. At each heartbeat, a volume of blood is accelerated by the heart toward the head along the ascending aorta. In response, the patient's body recoils in the opposite direction. This motion is detected by sensitive accelerometers attached to the table. (The table is then brought to rest and returned to equilibrium before the next heartbeat.)

A graph of acceleration of the table versus time, termed a ballistocardiogram, is generated. Based on these measurements, the acceleration of the blood ejected by the heart can be determined. Patients with low blood accelerations generally have weakened heart muscles.

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in unknown units.

Part A

At what time is the speed of the table (and hence the speed of the blood in the opposite direction) a maximum? Hint A.1 How to read the graph

The graph is acceleration versus time. Remember that velocity is the signed area under the acceleration curve. As long as the acceleration is positive, the speed is increasing. Once the acceleration becomes negative, the speed will

decrease back to zero. ANSWER: ₃

Correct

Catching the Dot

This applet shows a dot moving with constant acceleration.

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Change the values of the initial position, initial velocity, and the acceleration of the car so that its center follows the same path as the dot. What is the equation that describes the motion of the dot?

Hint A.1 How to approach the problem

Think about which quantity ( , , or ) can be determined most easily. Once you find that value as accurately as possible, decide which of the remaining two will be easiest to determine. Once you get to the third quantity, you may have to refine the other two slightly based on how close you are to matching the dot's trajectory.

Once you have determined the correct values, think about which kinematic formula gives the value of in terms of , , , and . Substitute the appropriate values into the formula and you will have the equation describing the trajectory of the dot.

Hint A.2 Find

Find the initial position of the dot. This should be relatively easy: Look at the scale and estimate where on the x-axis the dot is located. Move the slider so that the car is at the same point and then look at the picture to be sure that the center of the car is lined up with the dot.

Express your answer in meters to three significant figures. ANSWER:

= 12.0 Correct

Hint A.3 Find

Find the initial velocity of the dot. This will be much easier if you find first and set the car and dot to the same . Then, you may want to adjust the and settings for the car at the same time to help make the trajectory of the car more similar to that of the dot.

Hint A.3.1 Recognizing differences in speed and acceleration Hint not displayed Express your answer in meters per second to three significant figures.

ANSWER:

=-12.0 Answer Requested

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Find the acceleration of the dot. This will be much easier if you find first and set the car and dot to the same . You may want to adjust the and settings for the car at the same time to help make the trajectory of the car closer to that of the dot. It will be helpful to get the value of close to that of the dot before trying to match the accelerations.

Hint A.4.1 Recognizing differences in speed and acceleration Hint not displayed Express your answer in meters per second squared to two significant figures.

ANSWER:

= 2.0 Correct

Hint A.5 The appropriate equation

Recall that the kinematic equation describing the motion of an object with constant acceleration is the following: .

Express your answer in terms of . Enter all of the numbers in the equation without their units, as it is understood that you are using meters, meters per second, and meters per second squared

for distances, speeds, and accelerations, respectively. ANSWER:

=

All attempts used; correct answer displayed

Part B

The simulation runs until , even though the dot is far off of the screen by then. What is the position of the dot at time ?

Express your answer in meters to at least two significant figures. ANSWER:

= 172 Correct

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To take off from the ground, an airplane must reach a sufficiently high speed. The velocity required for the takeoff, the takeoff velocity, depends on several factors, including the weight of the aircraft and the wind velocity.

Part A

A plane accelerates from rest at a constant rate of 5.00 along a runway that is 1800 long. Assume that the plane reaches the required takeoff velocity at the end of the runway. What is the time needed to take off?

Hint A.2 Find the equation for the distance traveled by the plane

Hint not displayed Express your answer in seconds.

ANSWER:

= 26.83 Correct

Part B

What is the speed of the plane as it takes off? Hint B.1 How to approach the problem

Since you are given the constant acceleration of the plane, and you have also found the time it takes to take off, you can calculate the speed of the plane as it ascends into the air using the equation for the velocity of an object in motion at constant acceleration.

Hint B.2 Find the equation for the velocity of the plane

Which expression best describes the velocity of the plane after a certain interval of time ? Let be the initial velocity of the plane, and use for the acceleration of the plane. Remember that the plane starts from rest.

ANSWER:

=

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Alternatively, you can use the relation (recalling that in this case ).

Express your answer numerically at meters per second. ANSWER:

= 134 _Correct

Part C

What is the distance traveled by the plane in the first second of its run? Hint C.1 How to approach the problem

Apply the same equation that you used to solve Part A.

Express your answer numerically in meters. ANSWER:

= 2.50 _Correct

Part D

What is the distance traveled by the plane in the last second before taking off? Hint D.1 How to approach the problem

Hint not displayed Express your answer numerically in meters.

ANSWER:

= 132 Correct

Since the plane is accelerating, the average speed of the plane during the last second of its run is greater than its average speed during the first second of the run. Not surprisingly, so is the distance traveled.

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What percentage of the takeoff velocity did the plane gain when it reached the midpoint of the runway? Hint E.1 How to approach the problem

You need to find the velocity of the plane by the time it covers half the length of the runway and compare it with the takeoff velocity. Apply the same method that you used to determine the takeoff velocity.

Express your answer numerically. ANSWER: ₇₀

This is a "rule of thumb" generally used by pilots. Since the takeoff velocity for a particular aircraft can be computed before the flight, a pilot can determine whether the plane will successfully take off before the end of the runway by verifying that the plane has gained 70% of the takeoff velocity by the time it reaches half the length of the runway. If the plane hasn't reached that velocity, the pilot knows that there isn't enough time to reach the needed takeoff

velocity before the plane reaches the end of the runaway. At that point, applying the brakes and aborting the takeoff is the safest course of action.

Direction of Velocity and Acceleration Vector Quantities Conceptual Question

For each of the motions described below, determine the algebraic sign ( , , or ) of the velocity and acceleration of

the object at the time specified. For all of the motions, the positive y axis is upward. Part A

An elevator is moving downward when someone presses the emergency stop button. The elevator comes to rest a short time later. Give the signs for the velocity and the acceleration of the elevator after the button has been pressed but before the elevator has stopped.

Hint A.1 Algebraic sign of velocity

Hint A.2 Algebraic sign of acceleration

Enter the correct sign for the elevator's velocity and the correct sign for the elevator's

acceleration, separated by a comma. For example, if you think that the velocity is positive and the acceleration is negative, then you would enter +,- . If you think that both are zero, then you would enter 0,0 .

ANSWER: _-,+

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Part B

A child throws a baseball directly upward. What are the signs of the velocity and acceleration of the ball immediately after the ball leaves the child's hand?

Hint B.1 Algebraic sign of velocity

Hint B.2 Algebraic sign of acceleration

Enter the correct sign for the baseball's velocity and the correct sign for the baseball's

ANSWER: _+,-

Correct

Part C

A child throws a baseball directly upward. What are the signs of the velocity and acceleration of the ball at the very top of the ball's motion (i.e., the point of maximum height)?

Hint C.1 Algebraic sign of velocity

Hint C.2 Algebraic sign of acceleration

Enter the correct sign for the baseball's velocity and the correct sign for the baseball's

ANSWER: _0,-

Correct

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A stone is thrown upward from the edge of a cliff, reaches its maximum height, and then falls down into the valley below. A motion diagram for this situation is given, beginning the instant the stone leaves the thrower’s hand. Construct

the corresponding motion graphs taking the acceleration due to gravity as 10 . Ignore air resistance. In all three motion graphs, the unit of time is in seconds and the unit of displacement is in meters. In plotting the points, round-off the coordinate values to the nearest integer.

Part A

Construct a graph corresponding to the stone's vertical displacement, . Hint A.1 How to approach the problem

The motion diagram indicates the position of the stone at successive values of time, as well as the coordinate system being used to analyze the motion. By examining the first "dot" on the motion diagram, you can determine the initial value of the position of the stone, which is where your position graph should begin. By examining the last "dot" on the

motion diagram, you can determine the final position of the stone, which is where your position graph should end.

Hint A.2 Find the initial value of the stone's position

Is the initial value of position positive, negative, or zero? ANSWER: _positive

negative zero Correct

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Is the final value of position positive, negative, or zero? ANSWER: _positive negative zero Correct ANSWER: View

Part B

Construct a graph corresponding to the stone's vertical velocity, . Hint B.1 Find the initial value of the stone's velocity

Hint B.2 Find the final value of the stone's velocity

Hint not displayed ANSWER:

View Correct

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Part C

Construct a graph corresponding to the stone's vertical acceleration, . Hint C.1 How to approach the problem

View Correct

Motion of a Shadow

A small source of light is located at a distance from a vertical wall. An opaque object with a height of moves toward the wall with constant velocity of magnitude . At time , the object is located at the source .

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Find an expression for , the magnitude of the velocity of the top of the object's shadow, at time . Hint A.1 Calculate ratios of triangles

Hint A.2 Calculate the derivative

Hint A.3 _Find

Hint A.4 Substitute

Hint not displayed Express the speed of the top of the object's shadow in terms of , , , and .

ANSWER:

=

Correct

Overcoming a Head Start

Cars A and B are racing each other along the same straight road in the following manner: Car A has a head start and is a distance beyond the starting line at . The starting line is at . Car A travels at a constant speed . Car B starts at the starting line but has a better engine than Car A, and thus Car B travels at a constant speed , which is greater than .

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How long after Car B started the race will Car B catch up with Car A? Hint A.1 Consider the kinematics relation

Hint A.2 What is the relation between the positions of the two cars?

Hint A.3 Consider Car B's position as a function of time

Hint not displayed Express the time in terms of given quantities.

ANSWER:

=

Correct

Part B

How far from Car B's starting line will the cars be when Car B passes Car A? Hint B.1 Which expression should you use?

Hint not displayed Express your answer in terms of known quantities. (You may use as well.)

ANSWER:

=

Correct

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Learning Goal: To understand the relationships between position, velocity, and acceleration.

For this tutorial, use the PhET simulation The Moving Man. This simulation allows you to drag a person back and forth and look at the resulting position, velocity, and acceleration. You can also enter a position as a function of

time mathematically and look at the resulting motion.

Start the simulation. When you click the simulation link, you may be asked whether to run, open, or save the

file. Choose to run or open it.

You can click and drag the person left and right, or enter a numeric value in the boxes on the left panel to see plots for the person’s position, velocity, and acceleration as a function of time. Click Stop to stop the simulation. You can also watch a playback or click Clear on any of the graphs to start over. The grey bar can be dragged over the plot to any value in time, and the digital readouts will show the corresponding values of the position, velocity, and acceleration. The buttons on the right of each plot allow you to zoom in or out to resize the plots.

Under the Special Features menu, the Expression Evaluator option produces a second window in which you

can mathematically type in any function for the position as a function of time, . After typing in a function, click Go! to start the simulation.

To zoom in horizontally or vertically, click any of the three + buttons under the horizontal or vertical arrows.

Feel free to play around with the simulation. When you are done, click Clear on the main window before beginning Part A. Part A

First, you will focus on the relationship between velocity and position. Recall that velocity is the rate of change of position ( ). This means that the velocity is equal to the slope of the Position vs. Time graph.

Move the person to the position or enter –6.00 in the position box, and then click Clear (you do not need to click Stop first). Next, drag the person to the right to roughly and reverse his direction, returning him to the original position, at . Move the person relatively quickly, about a few seconds for the round trip. Your plots should look something like those in shown below.

Look at the Position vs. Time and Velocity vs. Time plots. What is the person's velocity when his position is at its maximum value (around 6 )?

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ANSWER:

The person's velocity is

zero. negative. positive. Correct

When the person’s position is a maximum, the slope of the position with respect to time is zero, so . However, due to the person’s acceleration, the velocity does not remain zero; he eventually moves to the left.

Part B

Acceleration is the rate of change of the velocity, , so it is the slope of the Velocity vs. Time graph. Because it is difficult to drag the person in a consistent and reproducible way, use the Expression Evaluator under the Special Features menu for this question.

Click Clear and type in the function in the Expression Evaluator. Press Go! and let

the simulation run roughly 5 simulation seconds before pressing Stop. Use the zoom buttons to adjust the plots so they fit in the screen. You should see a plot similar to what you got in the previous question, but much smoother.

Look at the Position vs. Time, Velocity vs. Time, and Acceleration vs. Time plots. Hint B.1 How to approach the problem

When the person is 8 to the right of the origin,

both the velocity and the acceleration are nonzero. the velocity is zero but the acceleration is positive. the velocity is zero but the acceleration is negative. both the velocity and the acceleration are zero. Correct

At , the person turns to go back in the opposite direction. His velocity is zero, but his acceleration is negative since the velocity is decreasing with time. This is similar to throwing a ball straight up into the air; at its highest point, the velocity is zero but the acceleration is still directed downward.

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Keep the function in the Expression Evaluator. What is the value of the person’s acceleration at ?

Hint C.1 How to approach the problem

=

Correct

This is an example of one-dimensional motion with constant acceleration. The position of an object undergoing this type of motion obeys the kinematic equation . In this case, the initial velocity is and the acceleration is (since ).

Part D

In the previous question, the person had an initial velocity of and a constant acceleration of . How would the maximum distance he travels to the right of the origin change if instead his initial velocity were doubled ( )?

Hint D.1 How to approach the problem

Hint not displayed ANSWER: _{The maximum distance would double.}

The maximum distance would increase by a factor of four. The maximum distance would not change.

Correct

Because it takes twice as much time to momentarily stop, and because his average velocity will be twice as fast, the distance he travels will be four times greater. Using the kinematic

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Part E

Now, assume that the position is given by the equation .

Enter this function in the Expression Evaluator as . Run the simulation for roughly three simulated seconds, and then look at the graphs. You will have to zoom in horizontally and zoom out vertically. Which of the following statements is true?

ANSWER: _{The acceleration is increasing at a constant rate.}

The position is increasing at a constant rate. The acceleration is constant in time.

The velocity is increasing at a constant rate. Correct

The graph showing Acceleration vs. Time is a straight line.

Part F

What is the position of the person when ?

Express your answer numerically in meters to one significant figure. ANSWER: ₄

Correct

Notice that since the position is given by , when the time is , the position is

Part G

What is the velocity of the person when ? Hint G.1 How to approach the problem

Express your answer numerically in meters per second to two significant figures. ANSWER: ₁₂

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time, is given by . So when , .

Part H

What is the acceleration of the person when ?

Express your answer numerically in meters per second squared to two significant figures. ANSWER: ₂₄

Correct

Notice that since the position is given by , the acceleration, which is the first derivative of velocity with respect to time, is given by . So when , . Notice also that the acceleration is proportional to time, which explains why it is increasing at a constant rate (as discovered in part G).

PhET Interactive Simulations University of Colorado http://phet.colorado.edu

Rearending Drag Racer

To demonstrate the tremendous acceleration of a top fuel drag racer, you attempt to run your car into the back of a dragster that is "burning out" at the red light before the start of a race. (Burning out means spinning the tires at high speed to heat the tread and make the rubber sticky.)

You drive at a constant speed of toward the stopped dragster, not slowing down in the face of the imminent collision. The dragster driver sees you coming but waits until the last instant to put down the hammer, accelerating from the starting line at constant acceleration, . Let the time at which the dragster starts to accelerate be .

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What is , the longest time after the dragster begins to accelerate that you can possibly run into the back of the dragster if you continue at your initial velocity?

Hint A.1 Calculate the velocity

=

Correct

Part B

Assuming that the dragster has started at the last instant possible (so your front bumper almost hits the rear of the dragster at ), find your distance from the dragster when he started. If you calculate positions on the way to this solution, choose coordinates so that the position of the drag car is 0 at . Remember that you are solving for a distance (which is a magnitude, and can never be negative), not a position (which can be negative).

Hint B.1 Drag car position at time

Hint B.2 Distance car travels until tmax

Hint B.3 Starting position of car

Hint B.4 Obtaining the Solution

=

Correct

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Find numerical values for and in seconds and meters for the (reasonable) values (26.8 m/s) and .

Separate your two numerical answers by commas, and give your answer to two significant figures. ANSWER:

, = 0.54,7.2 _Correct s, m

The blue curve shows how the car, initially at , continues at constant velocity (blue) and just barely touches the accelerating drag car (red) at .

Red Light, Green Light

A car and a train move together along straight, parallel paths with the same constant cruising speed . At the car driver notices a red light ahead and slows down with constant acceleration . Just as the car comes to a full stop, the light immediately turns green, and the car then accelerates back to its original speed with constant acceleration . During the same time interval, the train continues to travel at the constant speed .

Part A

How much time does it take for the car to come to a full stop?

Express your answer in terms of and ANSWER:

=

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Part B

How much time does it take for the car to accelerate from the full stop to its original cruising speed?

Express your answer in terms of and . ANSWER:

=

Correct

Part C

The train does not stop at the stoplight. How far behind the train is the car when the car reaches its original speed again? Hint C.1 How far did the train travel?

Hint C.2 How far did the car travel?

Express the separation distance in terms of and . Your answer should be positive. ANSWER:

=

Correct

Relative Velocity vs. Time Graph Ranking Task

Two cars travel on the parallel lanes of a two-lane road. The cars are at the same location at time , and move in such a way as to produce the velocity (relative to the ground) vs. time graph shown in the figure. On the graph, one

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vertical block is equivalent to one velocity unit.

Part A

Rank car #1’s speed relative to the ground at the lettered times (A through E). Hint A.1 Speed relative to the ground

The graph displays the velocity of each car relative to the ground. Since you are asked only about car #1, simply ignore car #2 and interpret the graph. Recall that speed is the magnitude of velocity.

Hint A.2 Find the speed of car #1 at time A

What is the speed of car #1 relative to the ground at time A? Remember that speed is the magnitude of velocity. ANSWER: _{greater than 1 unit}

between 0 and 1 unit less than 0 units Correct

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ANSWER:

View

Part B

Rank car #1’s speed relative to car #2 at the lettered times. Hint B.1 Relative speed

Hint B.2 Determine the directions of travel at time A

Hint B.3 Determine the directions of travel at time D

Hint not displayed Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

View Correct

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Rank the distance between the cars at the lettered times.

Hint C.1 Determine the directions of travel at time A

Hint C.2 When do the cars stop traveling in opposite directions?

Hint C.3 How is the separation distance changing?

Hint not displayed Rank from largest to smallest. To rank items as equivalent, overlap them.

ANSWER:

View Correct

Running and Walking

Tim and Rick both can run at speed and walk at speed , with . They set off together on a journey of distance . Rick walks half of the distance and runs the other half. Tim walks half of the time and runs the other half.

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How long does it take Rick to cover the distance ? Hint A.1 Compute midpoint for Rick

Hint A.2 Compute running time for Rick

Hint A.3 What equation to use

Hint not displayed Express the time taken by Rick in terms of , , and .

ANSWER:

=

Correct

Part B

Find Rick's average speed for covering the distance . Hint B.1 _{Calculate velocity using and time}

Hint not displayed Express Rick's average speed in terms of and .

ANSWER:

=

Correct

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How long does it take Tim to cover the distance? Hint C.1 Calculate average speed

Hint not displayed Express the time taken by Tim in terms of , , and .

ANSWER:

=

Correct

Part D

Who covers the distance more quickly?

Hint D.1 Consider the relative positions at the midpoint

Hint not displayed Think logically, but without using the detailed answers in the previous parts.

ANSWER: _Rick

Tim

Neither. They cover the distance in the same amount of time. Correct

Part E

In terms of given quantities, by what amount of time, , does Tim beat Rick?

It will help you check your answer if you simplify it algebraically and check the special case .

Express the difference in time, in terms of , , and . ANSWER:

=

Correct

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In the special case that , what would be Tim's margin of victory ? Hint F.1 Think it through

ANSWER: ₀

Correct

Speed of a Bullet

A bullet is shot through two cardboard disks attached a distance apart to a shaft turning with a rotational period , as shown.

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Derive a formula for the bullet speed in terms of , , and a measured angle between the position of the hole in the first disk and that of the hole in the second. If required, use , not its numeric equivalent. Both of the holes lie at the same radial distance from the shaft. measures the angular displacement between the two holes; for instance,

means that the holes are in a line and means that when one hole is up, the other is down. Assume that the bullet must travel through the set of disks within a single revolution.

Hint A.1 Consider hole positions

Hint A.2 _{How long does it take for the disks to rotate by an angle ?} Hint not displayed ANSWER:

=

Correct

The Graph of a Sports Car's Velocity

The graph in the figure shows the velocity of a sports car as a function of time . Use the graph to answer the following questions.

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Find the maximum velocity of the car during the ten-second interval depicted in the graph. Hint A.1 How to approach the problem

Hint not displayed Express your answer in meters per second to the nearest integer.

ANSWER:

= 55 Correct

Part B

During which time interval is the acceleration positive? Hint B.1 Finding acceleration from the graph

Hint not displayed Indicate the best answer.

ANSWER: _to to to to to Correct Part C

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Find the maximum acceleration of the car. Hint C.1 How to approach the problem

Hint C.2 Find the final velocity on the interval with greatest acceleration

Hint not displayed Express your answer in meters per second per second to the nearest integer.

ANSWER:

= 30 _Correct

Part D

Find the minimum magnitude of the acceleration of the car. Hint D.1 How to approach the problem

Hint not displayed Express your answer in meters per second per second to the nearest integer.

ANSWER:

= 0 Correct

Part E

Find the distance traveled by the car between and . Hint E.1 How to approach the problem

Hint E.2 Find the distance traveled in the first second

Hint E.3 Find the distance traveled in the second second

Hint not displayed Express your answer in meters to the nearest integer.

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ANSWER:

= 55 _Correct

Velocity from Graphs of Position versus Time

An object moves along the x axis during four separate trials. Graphs of position versus time for each trial are shown in the figure.

Part A

During which trial or trials is the object's velocity not constant?

Hint A.1 Finding velocity from a position versus time graph

Hint A.2 Equation for slope

Hint not displayed Check all that apply.

ANSWER: _{Trial A}

Trial B Trial C Trial D

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The graph of the motion during Trial B has a changing slope and therefore is not constant. The other trials all have graphs with constant slope and thus correspond to motion with constant velocity.

Part B

During which trial or trials does the object have the greatest average velocity? Hint B.1 Definition of average velocity

Hint not displayed Check all that apply.

ANSWER: _{Trial A}

Trial B Trial C Trial D Correct

You recognized that although the magnitudes of the average velocity in Trial B and Trial D are equal, their directions are opposite. This makes the average velocity in Trial D less than the average velocity in Trial B. The object does not move during Trial C, so it has an average velocity of zero. During Trial A the object has a positive average velocity but its magnitude is less than that in Trial B.

A Flea in Flight

In this problem, you will apply kinematic equations to a jumping flea. Take the magnitude of free-fall acceleration to be 9.80 . Ignore air resistance.

Part A

A flea jumps straight up to a maximum height of 0.540 . What is its initial velocity as it leaves the ground? Hint A.1 Finding the knowns and unknowns

Hint A.2 Determine which kinematic equation to use

Hint A.3 Some algebra help

Hint not displayed ✔

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Express your answer in meters per second to three significant figures. ANSWER: = 3.25 Correct Part B

How long is the flea in the air from the time it jumps to the time it hits the ground? Hint B.1 How to approach the problem

Hint B.2 Find the time from the ground to the flea's maximum height

Hint B.3 Find the time from the flea's maximum height to the ground

Hint not displayed Express your answer in seconds to three significant figures.

ANSWER:

time in air = 0.664_Correct

Notice that the time for the flea to rise to its maximum height is equal to the time it takes for it to fall from that height back to the ground. This is a general feature of projectile motion (any motion with ) when air resistance is neglected and the landing point is at the same height as the launch point.

There is also a way to find the total time in the air in one step: just use

and realize that you are looking for the value of for which .

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A motorcycle is following a car that is traveling at constant speed on a straight highway. Initially, the car and the

motorcycle are both traveling at the same speed of 43.0 , and the distance between them is 58.0 . After = 3.00 , the motorcycle starts to accelerate at a rate of 5.00 . The motorcycle catches up with the car at some time .

Part A

Which of the graphs correctly displays the positions of the motorcycle and car as functions of time?

Hint A.1 Describe the graph of the motorcycle's position

Hint A.2 The relative positions of the two vehicles

ANSWER: _a b c d e Correct Part B

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How long does it take from the moment when the motorcycle starts to accelerate until it catches up with the car? In other words, find .

Hint B.1 Using a moving reference frame

Hint B.2 Find the initial conditions for the position of the car

Hint B.3 Find the initial conditions for the position of the motorcycle

Hint B.4 Solving for the time

Hint not displayed Express the time numerically in seconds.

ANSWER:

= 4.82 _Correct s

Part C

How far does the motorcycle travel from the moment it starts to accelerate (at time ) until it catches up with the car (at time )?

Hint C.1 Find the initial conditions for the position of the motorcycle

Hint not displayed Answer numerically in meters.

ANSWER:

= 150 Correct m

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A rocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 58.8 . The acceleration period lasts for time 10.0 until the fuel is exhausted. After that, the rocket is in free fall.

Part A

Find the maximum height reached by the rocket. Ignore air resistance and assume a constant acceleration due to gravity equal to 9.8 .

Hint A.2 Find the height reached during the fueled part of the motion

Hint A.3 Find the initial velocity, the final velocity, and the acceleration for the "free-fall" part of the motion

Hint A.4 Determine which kinematic equation to use

Hint not displayed Write your answer numerically in units of meters.

ANSWER:

= 2.06×10_Correct 4

PSS 3.2 Relative Velocity

Learning Goal: To practice Problem-Solving Strategy 3.2 Relative Velocity.

An airplane pilot wishes to fly directly westward. According to the weather bureau, a wind of 75.0 is

blowing southward. The speed of the plane relative to the air (called the air speed) as measured by instruments aboard the plane is 310 . In which direction should the pilot head?

Problem-Solving Strategy: Relative velocity

IDENTIFY the relevant concepts:

Whenever you see the phrase “velocity relative to” or “velocity with respect to", it’s likely that the concepts of relative velocity will be helpful.

SET UP the problem using the following steps:

Label each frame of reference in the problem. Each moving body has its own frame of reference; in addition, you’ll almost always have to include the frame of reference of the earth’s surface. Use the labels to help identify the target variable.

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EXECUTE the solution as follows: Solve for the target variable:

.

Note the order of the double subscripts on the velocities: always means "the velocity of A (first subscript) relative to B (second subscript)". A useful rule for keeping the order of things straight is to regard each double subscript as a fraction. Then, the fraction on the left side is the product of the fractions on the right side. For a point P relative to frame

A, .

EVALUATE your answer:

Be on the lookout for stray minus signs in your answer. This may mean that you accidentally calculated the velocity of B relative to A instead of the velocity of A relative to B. If you have made this mistake, you can recover since .

IDENTIFY the relevant concepts

In the problem statement, you are given the speed of the plane relative to the air. Therefore, the concept of relative velocity will be used to solve for the pilot's direction.

SET UP the problem

Part A

The wind is blowing southward at 75.0 . In what frame of reference is this speed measured? ANSWER: _{The speed is measured relative to the ground.}

The speed is measured relative to the plane. The speed is measured relative to the air. Correct

The speed of the air is measured relative to a person standing on the ground, that is, it is measured in the frame of reference of the earth's surface. We will label this frame "G". Therefore, the wind speed is . In addition, you are told that the speed of the plane is measured relative to the air, so .

EXECUTE the solution as follows

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An airplane pilot wishes to fly directly westward. According to the weather bureau, a wind of 75.0 is

blowing southward. The speed of the plane relative to the air (called the air speed) as measured by instruments aboard the plane is 310 . In which direction should the pilot head?

Hint B.1 Determine the equation for the velocity of the plane relative to the ground

Hint B.2 Identify the orientation of the vectors

Hint B.3 Vector geometry

Express your answer in degrees to three significant figures. Take angles measured clockwise from the west to be positive.

ANSWER:

Direction =14.0 All attempts used; correct answer displayed

EVALUATE your answer

Part C

In this particular problem, you can compare the speed of the plane relative to the ground (the ground speed) to known quantities to evaluate the reasonableness of your answer. Find the ground speed of the plane, .

Hint C.1 The Pythagorean theorem

Hint not displayed Express your answer in kilometers per hour to three significant figures.

ANSWER:

= 301 _Correct

The SR-71 Blackbird, one of the fastest and highest flying airplanes currently in operation, can reach speeds of 2400 , and average commercial airliners, such as the Boeing 747, require speeds of about 290 to take off. Does the speed you calculated make sense when compared to these numbers?

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A Wild Ride

A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by

, where is a positive dimensionless constant.

Part A

At is the roller coaster car ascending or descending? Hint A.1 How to approach the problem

Hint A.2 Find the vertical component of the velocity of the car

Hint not displayed ANSWER: _ascending

descending Correct

Part B

Derive a general expression for the speed of the car. Hint B.1 How to approach the problem

Hint B.2 Magnitude of a vector

Hint B.3 Find the components of the velocity of the car

Hint not displayed Express your answer in meters per second in terms of and .

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ANSWER:

=

Correct

Part C

The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding . Find the maximum value of allowed by these regulations.

Hint C.1 How to approach the problem

Hint C.2 Find the maximum value of the speed

Hint not displayed Express your answer using two significant figures.

ANSWER:

= 1.7 Correct

Uniform Circular Motion

Learning Goal: To find the velocity and acceleration vectors for uniform circular motion and to recognize that

this acceleration is the centripetal acceleration.

Suppose that a particle's position is given by the following expression:

. Part A

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Choose the answer that best completes the following sentence: The particle's motion at can be described by ____________.

ANSWER: _{an ellipse starting at time} _{on the positive x axis}

an ellipse starting at time on the positive y axis

a circle starting at time on the positive x axis

a circle starting at time on the positive y axis

Correct

The quantity is defined to be the angular velocity of the particle. Note that must have units of radians per second. If is constant, the particle is said to undergo uniform circular motion.

Part B

When does the particle first cross the negative x axis?

Express your answer in terms of some or all of the variables , , and . ANSWER:

=

Correct

Now, consider the velocity and speed of the particle. Part C

Find the particle's velocity as a function of time. Hint C.1 _{Derivative of}

Express your answer using unit vectors (e.g., + , where and are functions of , , , and ). ANSWER:

=

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Find the speed of the particle at time .

Hint D.1 Definition of the magnitude of a vector

Hint D.2 Complete an mportant trig identity

Hint not displayed Express your answer in terms of some or all of the variables , , and .

ANSWER:

Correct

Note that the speed of the particle is constant: .

Part E

Now find the acceleration of the particle.

Express your answer using unit vectors (e.g., + , where and are functions of , , , and ). ANSWER:

=

Correct

Part F

Your calculation is actually a derivation of the centripetal acceleration. To see this, express the acceleration of the particle in terms of its position .

Express your answer in terms of some or all of the variables and . ANSWER:

=

Correct

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Now find the magnitude of the acceleration as a function of time.

Express your answer in terms of some or all of the variables , , and . ANSWER:

=

Correct

Part H

Finally, express the magnitude of the particle's acceleration in terms of and using the expression you obtained for the speed of the particle.

Express your answer in terms of one or both of the variables and . ANSWER:

=

There are three important things to remember about centripetal acceleration:

1. The centripetal acceleration is simply the acceleration of a particle going around in a circle. 2. It has magnitude of either or .

3. It is directed radially inward.

Velocity in a Moving Frame Conceptual Question

You are attempting to row across a stream in your rowboat. Your paddling speed relative to still

water is 3.0 (i.e., if you were to paddle in water without a current, you would move with a speed of 3.0 ). You head off by rowing directly north, across the stream.

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Part A

Draw the vector , representing your velocity relative to still water.

Draw the vector starting at the rowboat. The location, orientation and length of the vector will be graded. The vector's length is displayed in meters per second.

ANSWER:

View Correct

Part B

Now assume that the stream flows east at 4.0 . Draw the vectors , representing the velocity of the stream, and , representing the velocity of your rowboat relative to the stream bank. Be sure to draw both vectors.

Hint B.1 How to find the total velocity

Draw and starting at the tail and tip of respectively. The location, orientation and length of the vectors will be graded. Each vector's length is displayed in meters per second.

ANSWER:

View Correct

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Based on the vector diagram in Part B, determine how far downstream of your starting point you will finally reach the opposite shore if the stream is 6.0 meters wide.

Hint C.1 Find the time to cross the stream

= 8 _Correct

A Canoe on a River

A canoe has a velocity of 0.370 southeast relative to the earth. The canoe is on a river that is flowing at 0.600 east relative to the earth.

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Find the magnitude of the velocity of the canoe relative to the river. Hint A.1 How to approach the problem

Hint A.2 Find the relative velocity vector

Hint A.3 Find the components of the velocity of the canoe relative to the river

Hint not displayed Express your answer in meters per second.

ANSWER:

= 0.428 _Correct

Part B

Find the direction of the velocity of the canoe relative to the river. Hint B.1 How to approach the problem

Hint B.2 Find the direction of a vector given its components

Consider a vector of magnitude whose x component is and y component is . What is the angle this vector makes with the x axis?

Hint B.2.1 The direction of a vector

ANSWER:

(54)

Express your answer as an angle measured south of west. ANSWER: _37.7

degrees south of west

Curved Motion Diagram

The motion diagram shown in the figure represents a pendulum released from rest at an angle of 45 from the vertical. The dots in the motion diagram represent the positions of the pendulum bob at eleven moments separated by equal time intervals. The green arrows represent the average velocity between adjacent dots. Also given

is a "compass rose" in which directions are labeled with the letters of the alphabet.

Part A

What is the direction of the acceleration of the object at moment 5? Hint A.1 How to approach the problem

Hint A.2 Definition of acceleration

Hint A.3 Change of velocity: a graphical interpretation

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ANSWER: _A

Correct

Part B

What is the direction of the acceleration of the object at moments 0 and 10? Hint B.1 Find the direction of the velocity

Hint B.2 Definition of acceleration

Hint B.3 Applying the definition of acceleration

Enter the letters corresponding to the arrows with these directions from the compass rose in the figure, separated by commas. Type Z if the acceleration vector has zero length.

ANSWER: