Chapter 2 Kinematics in One Dimension
2.1 Displacement
Kinematics deals with the concepts that are needed to describe motion without any
reference to forces.
Dynamics deals with the effect that forces have on motion.
Together, kinematics and dynamics form the branch of physics know as mechanics.
Definition of Displacement
The displacement is a vector that points from an object’s initial position to its final
position and has a magnitude that equals the shortest distance between the two
positions.
SI Unit of Displacement: meter (m)
Displacement is a vector quantity that is both a magnitude and a direction. The
displacement can be related to
and by
We will often deal with motion along a straight line. In this case, a displacement in one
direction along the line is assigned a positive value, and a displacement in the opposite
direction is assigned a negative value. While you can use the +x direction as positive, this
really depends on the person doing the problem.
2.2 Speed and Velocity
Average speed is the distance traveled divided by the time required to cover the distance.
Average speed however, is only a magnitude and does not tell anything about the
direction being traveled.
Average Velocity
Definition of Average Velocity
̅
SI Unit of Average Velocity: meter per second (m/s)
Instantaneous Velocity
The instantaneous velocity V of an object indicated how fast the object moves and the
direction of the motion at each instant of time. The magnitude of the instantaneous
velocity is called the instantaneous speed, and it is the number (with units) indicated by
how fast something travels.
Sections 2.1 & 2.2
Practice Problems: 2, 6, 8
Homework Problems: 1, 3-5, 7
2.3 Acceleration
Acceleration occurs any time there is a change in speed or a change in direction.
Definition of Average Acceleration
Average acceleration – defined as the change in a velocity divided by the time it takes
for the change to occur.
̅
SI unit of Average Acceleration: meter per second squared (m/s2)
The average acceleration ̅ is a vector that points in the same direction as
in velocity.
, the change
Instantaneous acceleration – is the acceleration of an object at any give instant.
When using the term acceleration we usually are referring to the instantaneous
acceleration.
Whenever the acceleration and velocity vectors have opposite directions, the object
slows down and is said to be “decelerating”.
Sections 2.3
Practice Problems: 14, 15
Homework Problems: 12, 13, 16
2.4 Equations of Kinematics for Constant Acceleration
Consider an object to have an initial velocity
at time
and moves for a time t
with a constant acceleration a. The final velocity v can be determined using:
̅
or
(Constant acceleration)
Where
̅
̅
Because the acceleration is constant, the velocity increases at a constant rate.
So the average velocity ̅ is midway between the initial and final velocities is given as:
̅
(constant acceleration)
This applies only if the acceleration is constant and cannot be used when the acceleration
is changing, for this we would need calculus. The displacement at time t can now be
determined as:
̅
(constant acceleration)
In the previous equations there are five kinematic variables.
Displacement
̅
Acceleration (constant)
Final velocity at time t
Initial velocity at time t0=0 s
Time elapsed since t0=0 s
Table 2.1 – Table of Equations of Kinematics for Constant Acceleration
Equation #
Equation

(2.4)
(2.7)

(2.8)


(2.9)












2.5 Applications of the Equations of Kinematics
As long as the acceleration of an object remains constant, we are able to use the
kinematic equations.
Things to consider when working out problems.
1. Decide at the start which directions are to be called positive (+) and negative (-)
relative to a conveniently chosen coordinate origin.
a. While the decision is arbitrary, it must be made and maintained because the
displacement, velocity, and acceleration are vectors and their directions
must always be taken into consideration.
2. As you reason through a problem (before solving it) you need to interpret
“decelerating” or “deceleration” correctly, should they occur in the problem.
a. You need to determine from your velocity direction what the direction of the
acceleration is.
b. A decelerating object does not necessarily have a negative acceleration.
3. Sometimes there are two possible answers to a kinematic problem, each answer
corresponding to a different situation.
4. The motion of two objects may be interrelated, so they share a common variable.
The fact that the motions are interrelated is an important piece of information. In
such cases, data for only two variables need be specified for each object.
5. Often the motion of an object is divided into segments, each with a different
acceleration. When solving such problems, it is important to realize that the final
velocity for one segment is the initial velocity for the next segment.
Sections 2.4 & 2.5
Practice Problems: 21, 25, 27
Homework Problems: 19, 20, 22-24, 30
2.6 Freely Falling Bodies
When considering falling objects we will assume that the fall is relatively small compared
to the radius of the earth, and that there is no air resistance (unless specified). These
assumptions allow us to work with a constant acceleration therefore the kinematic
equations can be used when dealing with free-fall.
Acceleration due to gravity – The acceleration of a freely falling body.
The magnitude of the acceleration due to gravity is denoted by g. The direction of g is
directed downward, toward the center of the earth. Near the earth’s surface g is
approximately
Things to know about Free-fall/gravity:
 In reality g changes depending on the location on earth’s surface.
 In free-fall, all things fall with the same acceleration due to the surface gravity in
the ABSENCE of air resistance.
 The moon’s gravity is approximately one-sixth of the earth’s.
When using the kinematic equations for free-fall we typically change the displacement
variable to y (since y indicates height). The acceleration due to gravity is always a
downward-pointing vector.
Objects in free-fall are considered to be moving without any other assistance other than
gravity. These objects can be moving up or down (thrown or dropped).
The motion of an object that is thrown upward and eventually returns to earth contains a
symmetry that is useful to keep in mind from the point of view of problem solving.
 The time it takes to reach a maximum height is the same time it takes to return to the
starting point.
 A type of symmetry involving the speed also exists. An object thrown up will have the
same speed at the same position on its return trip down.
Sections 2.6
Practice Problems: 37, 44, 47
Homework Problems: 38-43, 45, 46
2.7 Graphical Analysis of Velocity and Acceleration
When working with position, velocity and acceleration graphs, remember:
 Position-Time graphs
o Slope is the velocity
o The tangent line is also the slope/velocity at a given point
o Shapes
 If curved, velocity is a linear graph, acceleration is a given value.
 If linear with a slope, velocity is constant, acceleration is zero.
 If no slope (constant), velocity is zero.
 Velocity-Time graphs
o Slope is the acceleration
o The tangent line is also the slope/acceleration at a given point (this should
be constant in this class).
o The area under the graph is the distance traveled by the object for that given
time interval.
o Shape
 If has a slope, position graph is curved, acceleration is a given value.
 If no slope (constant), position graph is linear graph with velocity as
the slope, no acceleration.
 If at zero, object is motionless.
 Acceleration-Time graphs
o In this class the slope should always be constant.
 If constant value, velocity graph is a linear graph with acceleration as
the slope, position graph is a curved graph.
 If zero, then object is moving with constant speed.
Sections 2.7
Practice Problems: 58, 61
Homework Problems: 57, 59, 60