Chapter 2: Linear Motion
Objectives
1. Understand that motion is relative
2. Define speed, velocity, acceleration, and understand the distinctions among them 3. Describe motion in free fall
4. Describe motion of an object thrown straight up and allowed to fall until it hits the ground 5. Determine the speed and distance fallen at any time after an object is dropped from rest, when
air resistance is negligible
6. Explain how graphs can be used to describe relationships among time, distance, and speed
Scalar: quantity with magnitude, but no direction Ex—time, mass, speed
Vector: quantity with magnitude and direction Ex—velocity, force
Motion is described relative to something
Everything moves. Even things that look like they’re at rest move relative to something
else.
Ex—you may be sitting still, but you are moving at about 30 km/s relative to the
Sun
Unless otherwise stated, assume we mean speed with respect to the surface of the
Earth.
Speed (scalar) is a measure of which distance is covered, and it is measure in units of distance divided by time
Speed = Distance_ Ex—m/s ; mi/hr ; cm/ms
Time
Ex—If you ride a bike a distance of 5m in 1s, what is your speed? 10m in 2s? 100m in 20s?
Instantaneous Speed is the speed at any instant during a time interval. Average Speed is the total distance covered divided by the time interval.
Average Speed = Total Distance_ Time Interval Velocity (vector) is speed together with direction.
Velocity is constant ONLY when speed and direction are BOTH constant.
o Ex—A car on a circular track may have a constant speed but not a constant velocity,
o Ex—The speedometer of a car moving northward reads 60 km/hr. It passes another car that travels southward at 60 km/hr. Do both cars have the same speed? Do they have the same velocity?
Changing Velocity: If either the speed or the direction (or both) is changing, then the velocity
is changing. Constant speed and constant velocity are NOT the same thing.
Acceleration (vector) is the rate at which velocity is changing with respect to time.
Acceleration = _Change of Velocity_ Ex—m/s/s ; km/hr^2
Time Interval
Acceleration is ANY change in speed, direction, or both! An object accelerates when its speed is
decreasing, when its speed is increasing and/or when its direction is changing.
o Ex—If you ride around a curve at a constant speed of 50 km/hr, you feel the effects of
acceleration as you body tends to move outward toward the outside of the curve. You may round the curve at constant speed, but your velocity is not constant, because your direction is changing every instant. The change in the state of motion is acceleration.
For now, we are only concerned with motion along a straight line.
The units of acceleration are a bit more complicated. Since acceleration is the change in velocity
per time interval, its units are those of Speed per Time with Direction.
o Ex—If you’re standing on a curb at 0 ft/s and then cross the street at 4 ft/s in 8 seconds,
your acceleration is:
Acceleration = _ Velocity _ = _Vfinal - Vinitial_ = _(4 ft/s – 0 ft/s)_ = 0.5 ft/s2
Time tfinal - tinitial (8 s – 0 s)
*Note that a unit for time enters twice: once for the unit of speed and again for the interval of time in which the speed is changing.
An object in free fall is falling under the influence of gravity alone when air resistance does not affect its motion.
In real life, air resistance affects the acceleration of a falling object. However, the presence of
air resistance in the math of a physics problem is extremely complicated. It is a generally accepted practice to ignore air resistance and assume that only gravity is affecting the falling object.
The acceleration of an object falling under conditions where air resistance is negligible is about
10 meters per second squared (10 m/s2). In cases where accuracy is more important, use the
value of 9.8 m/s2.
For free fall, it is customary to use the letter g to represent the acceleration because the
acceleration is due to gravity. In all other cases, it is customary to use the letter a to denote
The instantaneous velocity of an object falling from rest is equal to the acceleration multiplied by the amount of time it falls, the elapsed time (time interval).
Instantaneous velocity = acceleration x elapsed time
The instantaneous velocity v of an object falling from rest after an elapsed time t can be
expressed in equation form:
v = gt
where: v velocity
g acceleration due to gravity (10 m/s2)
t elapsed time
When an object is thrown straight upwards and allowed to free fall to the ground, it undergoes
Remember! Speed and Distance are not the same thing! How fast something moves is entirely different from how far it moves.
o Ex—Consider an object in free fall. It starts from rest at t = 0s, v = 0m/s. It falls for 4
seconds total, and accelerates at an average of 10 m/s2. By the time the clock reads t =
1s, the object has an instantaneous speed of 10 m/s. Does this mean it falls a distance
of 10 meters during this first second? NO!
If the object falls 10 meters the first second, it average speed is 10 m/s. But we
know the speed began at zero and took a full second to get to 10 m/s. So the average speed is between zero and 10 m/s. For any object moving in a straight line with constant acceleration, we find the average velocity the way we find the average of any two numbers: add them and divide by 2. So adding the initial velocity of zero and the final velocity of 10 m/s and then dividing by 2, we get 5 m/s. So the object falls a distance of 5 meters in that first second.
Average Velocity = _Vinitial + Vfinal_ = _(0m/s + 10 m/s)_ = 5 m/s
2 2
How far does the object travel during the entire 4 seconds of its free fall?
Distance = ________________ meters
d = vt only holds when v is the average speed or velocity.
Whenever an object’s initial speed is zero and the acceleration a is constant, the equations for
the velocity and distance traveled are:
v = at and d = ½ at2
If the object has an initial velocity, vinitial, some thought will show that the equations for velocity
and distance traveled become:
We can also use graphs to demonstrate an objects motion and the relationships between position and time, velocity and time, and acceleration and time.
Slope is the vertical change divided by the horizontal change for any given part of the line.
o The slope of the Position-Time Graph will give you the object’s corresponding Velocity.
o The slope of the Velocity-Time Graph will give you the object’s corresponding
Acceleration.
Ex—From time t = 0s to about t = 4s, the slope of the Position-Time (P-T) Graph
is positive and constant, therefore the Velocity is constant. Because the Velocity is constant, the Acceleration is constant.
Ex—From time t = 5s to t = 7s, the slope of the P-T Graph is negative and
constant, therefore the Velocity is constant.
Ex—From time t = 8s to t = 10s, the slope of the P-T Graph is positive and
inconstant, therefore, the Velocity is positive and inconstant.
Ex—From time t = 10s to t = 12s, the slope of the P-T graph is zero, therefore,
the Velocity and Acceleration are also zero.
*Note: the Zero crosses on the P-T Graph do not mean that the Velocity and
Acceleration are zero as well. This is only giving you position from your starting point.
Equations Used: v = d/t a = v/t v = gt v = (vi + vf)/2
d = ½ at2 v = v