CHAPTER 1 – KINEMATICS

• The study of motion is known as kinematics, from the Greek “kinema” (meaning motion or movement).

• In all high school physics kinematics we will be treating moving objects as point-like, i.e. we will not be concerned with the
dimensions of the object when determining how it is moving.

• The reality is much more complex – for objects that are not point-like, we must take into account how the mass is distributed
about the object’s centre of mass. If the object is spinning this further complicates things.

• Of particular importance is the analysis of bodies moving in two dimensions. For example, a circus performer launched from a
cannon undergoes two dimensional motion:

Ch. 1.1 - Kinematics 1

 PROJECTILE MOTION
• After discussing and observing the Geogebra animation of a particle undergoing the same motion, answer the following
questions:

1. What direction is the acceleration on the ball during launch? As it is moving through the air?

2. What quantities of the motion are constant and which change throughout out the flight of the ball?

3. Describe how the motion of the ball would appear if observed from the following locations:

a) From behind the cannon.

b) From above the cannon (i.e. looking straight down on the ball’s path).

Answers
1. Acceleration is on the same angle relative to the ground as the cannon during launch. Once in flight, the only acceleration acting on the ball is straight down (i.e. acceleration
due to gravity).
2. The horizontal component of the velocity is constant throughout (neglecting wind resistance). The vertical component of the velocity changes throughout the flight – it is +ve
before the max. height, zero at max. height, and –ve after max. height.
3. a) The ball’s motion would all appear up and down. It would start with high speed, slow until it reaches maximum height where it stops. Then it drops, increasing in speed until
it hits the ground.
3. b) From above the ball would appear to be moving in a straight line with constant velocity.

Ch. 1.1 - Kinematics 2

 CH. 1.1 – MOTION AND MOTION GRAPHS

Kinematics Terminology

• We have defined kinematics, however students often confuse this with dynamics. Kinematics is the study of motion with no
consideration for the forces responsible for the motion. Dynamics is concerned with the cause of motion, specifically the role
forces play in motion.

• All physical quantities can be collected into two groups, scalars and vectors

• Scalars are quantities that have only a size, or magnitude.

• Vectors are quantities require both a magnitude and direction to be fully described.

• There are many examples of both in physics:

Scalars Vectors

Distance Displacement
Speed Velocity
Energy Acceleration
Mass Force
Time

Ch. 1.1 - Kinematics 3

 CH. 1.1 – MOTION AND MOTION GRAPHS

• The simplest type of motion is movement along a single line, or one-dimensional motion. The diagram and plots below (from
pg. 8 of the text) illustrate the motion of hockey puck after being passed.

• The ice can be treated as frictionless, so the puck moves in a straight line with constant velocity, what is referred to as uniform
motion. The dots on the diagram in (a) are equally spaced to reflect this type of motion.

• The graph in (b) is a position-time graph, which plots the position of the puck as a function of time. Position is measured relative
to some reference point or origin and thus always has distance and direction (here direction is simply left or right).

• The change in the puck’s position is called its displacement, which is written as: Δ𝑑⃗ 𝑑⃗ 𝑑⃗ where 𝑑⃗ is the initial position
and 𝑑⃗ is the final position.

• The graph in (c) is a velocity-time graph, which can be constructed from the position-time graph. The slope of a line on the
position-time graph gives the velocity of the puck (which has the same direction as the displacement).

Ch. 1.1 - Kinematics 4

 SPEED AND VELOCITY

• In every-day language many people use the terms speed and velocity interchangeably, however there is a significant difference.

• Speed is a scalar quantity – it tells us how fast an object is moving, is always positive or zero, and requires no direction.

• The average speed is defined as the total distance covered divided by the total time to cover that distance:

Δ𝑑
𝑣
Δ𝑡

• Velocity is a vector quantity – it tells us how fast an object is moving in a specific direction, can be positive, negative, or zero,
and must have a direction (otherwise it is just a scalar). An arrow over the letter indicates the vector nature (𝑣⃗).

• The average velocity has a very similar formula, however it is a vector relationship:

Δ𝑑⃗ 𝑑⃗ 𝑑⃗
𝑣⃗
Δ𝑡 𝑡 𝑡

• When an object undergoes uniform motion, its average velocity is constant in both size
and direction throughout the motion. This will give a constant slope on the position-time
graph. In general, the average velocity can be found by connecting any two points on the
curve (which is a secant) and finding the slope.

Ch. 1.1 - Kinematics 5

 EXAMPLE – PG 10 Q1

A woman leaves her house to walk her dog. They stop a few times along a straight path. They walk a distance of 1.2 km [E] from
their house in 24 min. In another 24 min, they turn around and take the same path home.

a) Determine the average speed of the woman and her dog for the entire route.
b) Calculate the average velocity from their house to the farthest position from the house.
c) Calculate the average velocity for the entire route.
d) Explain why the answers to part b) and c) differ.

Ch. 1.1 - Kinematics 6

 INTERPRETING VELOCITY ON A GRAPH

• Our example of the hockey puck assumed uniform motion, where the position-time graph yielded a straight line with constant
slope (and thus a horizontal line on the velocity-time graph).

• If an object starts at rest and speeds up then the position-time graph will not be linear. The nature of the graph will largely
depend on the nature of the acceleration, however most of our p-t graphs will have the shape of a parabola.

• For example, after coming to rest at a stop sign, a car starts to speed up. The first graph below shows the p-t graph for the
motion, assuming the car starts at rest. The dots are all spaced at equivalent time intervals while the position is growing
between each pair which tells us the velocity is increasing.

• If we were to look at our speedometer during this motion, we would obtain a velocity-time graph similar to the one below.

Ch. 1.1 - Kinematics 7

 INSTANTANEOUS VELOCITY AND SPEED

• If we wish to know the velocity at a particular time (instead of over an interval) we wish to know the instantaneous velocity 𝑣⃗ at
that time. For example, the graph below left shows a p-t curve with a line connecting two points at 𝑡 1 s and 𝑡 2 s. This
line is known as a secant, and will tell us the average velocity of the object over this interval.

• If we want to know the instantaneous velocity at say, 𝑡 1.5 s, then we must allow these two times to approach each other until
the size of the interval becomes very small. When this occurs the secant becomes a tangent at that point.

• The graph below right shows the process of allowing the time intervals to shrink, turning our secant into a tangent line. The
slope of this tangent is our instantaneous velocity.

Here we make the time interval smaller

If we create an interval, such as and smaller until the secant line is a very
𝑡 1.0 s to 𝑡 2.0 s, and close approximation to the tangent line
connect these points we form a at the point 𝑡 1.5 s.
secant. The slope of this secant
gives us the average velocity over The slope of the tangent line gives us
this interval. the instantaneous velocity at the point
𝑡 1.5 s.

Ch. 1.1 - Kinematics 8

 INSTANTANEOUS VELOCITY AND SPEED

• If we simply take the magnitude of the slope of the tangent, then we obtain the instantaneous speed, 𝑣. This is just the speed of
the object at any given time.

• Think of instantaneous speed as the speedometer reading on a car, while instantaneous velocity also adds in the direction of
motion of the car.

• For short duration motion the instantaneous speed will often differ from the instantaneous velocity. However, for long trips the
two can be very similar. For periods of uniform motion they will be identical in magnitude.

EXAMPLE – PG. 12 SAMPLE PROBLEM 1

The position-time graph at right shows the details of how an object moved.

a) Calculate the average velocity during the time interval 𝑡 1.0 s to 𝑡 2.5 s.
b) Analyze the p-t graph to sketch a qualitative v-t graph of the object’s motion.

Ch. 1.1 - Kinematics 9

 ACCELERATION

• If a particle experiences a change in velocity or speed it is said to be undergoing an acceleration. Acceleration is a measure of
how velocity changes with respect to time. The SI unit of velocity is m/s, therefore our SI unit of acceleration is m/s .

• We can determine acceleration from a velocity-time graph. There are only two scenarios we will see in this course:

1. Uniform Motion

- An object is moving with constant velocity (or is stationary).

- The p-t graph will be a straight line with a non-zero slope.
- The v-t graph will be straight line with zero slope (so zero acceleration).

2. Uniform Acceleration

- An object is moving with constant acceleration.

- The p-t graph will be curved.
- The v-t graph will be a straight line with non-zero slope.

• When the velocity of an object changes by Δ𝑣⃗ over a time interval of Δ𝑡, we define the average acceleration 𝑎⃗ by the following
expression:
Δ𝑣⃗
𝑎⃗
Δ𝑡
• We can also define instantaneous acceleration, the acceleration at a particular instant in time; however, non-uniform acceleration
is highly complex and seldom dealt with, even at a university level.
Ch. 1.1 - Kinematics 10
 PRACTICE – TEXT BOOK PG 15

1. The figure at top right shows a graph of the motion of a car along a straight road.
a) How can you tell from the graph that the car has a constant acceleration?
b) Describe the motion of the car.
c) Determine the acceleration of the car.

2. Examine the velocity-time graph shown at bottom right.

a) Determine the average acceleration for the entire trip.
b) Determine the instantaneous acceleration at 3 s and at 5 s.
c) Draw a reasonable acceleration-time graph of the motion.

Ch. 1.1 - Kinematics 11