Ch 2.ppsx

Kinematics

the description of motion

(2.1) Motion along a Straight Line

average velocity, v_av-x – displacement divided by time interval

v_av-x = x₂ – x₁ t₂ – t₁

= x

t

displacement, x – change in position represented by a vector from P₁ to P₂

ex. x₁ = 19 m x₂ = 277 m

x-t graphs

• slope of line connecting P₁ and P₂ = v_av-x

(2.2) Instantaneous Velocity

• _{limit of average velocity as time interval shrinks to zero}

v_x = lim

t0 t

x _dx dt

=

derivative of position with respect to time

*on x-t graph, instantaneous velocity at any point = slope

Example 2.1

a) Find the cheetah’s displacement during the time interval from t₁= 1.0 s and t₂ = 2.0 s

b) Find the average velocity.

c) Find the instantaneous velocity at t = 1.0 and 2.0 s

Derive an expression for the instantaneous velocity as a function of time.

Review of x-t and v

-t graphs

x-t graph:

_________________ indicates object’s position

_________________ indicates object’s speed

_________________ indicates object’s direction

steepness of slope value on x-axis

Review of x-t and v

-t graphs

v_x-t graph:

_________________ indicates object’s speed

_________________ indicates object’s direction

value on v_x-axis

sign of v

(2.3) Average and Instantaneous

Acceleration

• _{acceleration describes the rate of change of}

velocity with time

a_av-x = v_2x – v_1x t₂ – t₁

= v_x

• _{instantaneous acceleration is the limit of average}

acceleration as time interval shrinks to zero

a_x = lim

t0 t

v_x dv_x dt

v

-t graphs

*on v_x-t graph, instantaneous

acceleration at any point = slope of tangent to

Example 2.3

v_x = 60 m/s + (0.5 m/s3)t2

a) Find the average acceleration between t₁ =1.0 s and t₂ = 3.0 s.

Relationship between a_x and x

dv_x dt

a_x = _{= dt}d _dtdx _{= dt}d2x₂

• a_x is the second derivative of x

• _{second derivative is directly related to the}

Concavity of

x-t graph

a

v

concave up positive increasing

concave

down negative decreasing

no concavity zero constant

x

v

x

a

x

d

er

iv

at

iv

e

an

tid

(2.6) Velocity and Position by

Integration

On trans-Atlantic flights an accelerometer is used

to measure the plane’s acceleration. From these

measurements, the

velocity and position may be found.

Graphical Approach to Integrals

• since a_av-x = v_x/t, we

can write:

v_x = a_av-xt

• _{but that can be represented}

by the area of these green rectangles

• _{in the limit that} _t_{0, the rectangles become very}

skinny and . . .

• we can find the change in v_x during some time

interval by summing the areas of all the skinny rectangles in that interval

• _{this process is called integration, and it’s}

equivalent to finding the area under the a_x-t

curve



dv

=



a

dt

v_2x

v_1x

t₂

t₁



v

- v



a

dt

t₂

• by repeating this procedure with the v_x-t graph, we

can find the change in x during some time interval

x

– x



v

dt

t₂

t₁

• if we choose t₁ = 0 and t₂ any later time, and if

we call the initial position x₀ and the initial velocity v_0x, then we have:

x

x



v

dt

t

0

v

v



a

dt

t

x(t)

x



v

dt

t

0

v

(t)

v



a

dt

t

0

Example 2.9

At t = 0, when Sally is driving her 1965 Mustang at 10 m/s she passes a signpost at x = 50 m. Her

acceleration is given by:

a_x = 2.0 m/s2 – (0.10 m/s3) t

a) Derive expressions for her velocity and position as functions of time.

Describe Sally’s motion, then graph a vs. t

Graph v vs. t and x vs. t.

Example 2.9

At t = 0, when Sally is driving her 1965 Mustang at 10 m/s she passes a signpost at x = 50 m. Her

acceleration is given by:

a_x = 2.0 m/s2 – (0.10 m/s3) t

b) At what time is her velocity greatest?

c) What is her maximum velocity?

(2.4) Motion with Constant

Acceleration

• _{Let’s use a calculus-based approach (just to be} different from last year)

• If we assume that a_x(t) = a_x, then:

v_x =

x =

v_0x + a_xt

• _{by combining these two, we get another useful}

(and familiar, I hope) equation:

v_x2 =

Try problem 1.13 from ActivPhysics Online

“Car Catches Truck”

www.aw.com/young11

v_0x2 + 2a

Problem 2.61

Assume the light turns yellow right now and stays yellow for 3.0 s. If the driver steps on the brake, the car slows at -3.7 m/s2. If he instead

steps on the gas pedal, the car accelerates at 3.2 m/s2. To avoid being in the intersection at

the instant that the light turns red, should the driver step on the brake or the gas?

(2.5) Free Fall

• _{use the equations for constant acceleration, but}

in the y direction

• let a_y = -g = - 9.8 m/s2

v_y = v0y + ayt

y = y₀ +v_0yt +1/2 a_yt2

v_y2 = v

Sketch y, v_y, and a_y vs. t graphs for a ball thrown upwards at 5 m/s.

y

t

v_y

t

a_y

Free Fall formulas?

v_y = v0y + ayt

y = y₀ +v_0yt +1/2 a_yt2

v_y2 = v

Try problem 1.7 from ActivPhysics Online

“Balloonist Drops Lemonade”

www.aw.com/young11