Kinematics: Describing Motion
Understand how physicists describe motion using displacement, velocity, acceleration and kinematic equations -- the mathematical foundation of mechanics.
Displacement, Velocity and Acceleration
In physics, we distinguish between scalar quantities (magnitude only) and vector quantities (magnitude and direction). Kinematics uses precise vector quantities to describe how objects move.
Displacement
The change in position of an object from its starting point. It is a vector -- it has both magnitude and direction. Measured in metres (m).
Velocity
The rate of change of displacement with respect to time. A vector quantity. Measured in metres per second (m/s or m s-1).
Acceleration
The rate of change of velocity with respect to time. A vector quantity. Measured in m/s2 (m s-2).
Scalar vs Vector Comparison
Scalar (magnitude only)
- Distance -- total path length
- Speed -- rate of distance change
- Time, mass, energy
Vector (magnitude + direction)
- Displacement -- straight-line change in position
- Velocity -- rate of displacement change
- Acceleration, force, momentum
The Kinematic Equations
For uniform (constant) acceleration, we use four key equations that relate displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t). These are sometimes called the SUVAT equations.
The Four Kinematic Equations
v = u + at
Final velocity from initial velocity and acceleration
s = ut + ½at2
Displacement from initial velocity and acceleration
v2 = u2 + 2as
Final velocity without time
s = ½(u + v)t
Displacement from average velocity
Key convention: Choose a positive direction (e.g. to the right or upward). Quantities in the opposite direction are negative. For projectiles near Earth's surface, use g = 9.8 m s-2 downward.
Motion Graphs
Graphs are essential tools for analysing motion. The two most important graphs in kinematics are displacement-time and velocity-time graphs.
Displacement-Time Graph
- Gradient = velocity at that instant
- Straight line = constant velocity
- Curve = changing velocity (acceleration)
Velocity-Time Graph
- Gradient = acceleration
- Area under curve = displacement
- Horizontal line = constant velocity (a = 0)
Connecting the Graphs
s-t graph
Gradient gives velocity
v-t graph
Gradient gives acceleration; area gives displacement
a-t graph
Area gives change in velocity
Key Vocabulary
Displacement
The straight-line distance and direction from an object's initial position to its final position. A vector quantity measured in metres (m).
Instantaneous Velocity
The velocity of an object at a specific instant in time, found from the gradient of the displacement-time graph at that point.
Uniform Acceleration
Acceleration that remains constant over time. The kinematic (SUVAT) equations apply only when acceleration is uniform.
Average Velocity
Total displacement divided by total time. Differs from average speed, which uses total distance instead.
Worked Examples
A car accelerates from rest at 2.5 m s-2 for 8.0 s. Find the final velocity and displacement.
Known: u = 0 m s-1, a = 2.5 m s-2, t = 8.0 s
Final velocity: v = u + at = 0 + 2.5 × 8.0 = 20 m s-1
Displacement: s = ut + ½at2 = 0 + ½ × 2.5 × 8.02 = ½ × 2.5 × 64 = 80 m
A ball is thrown upward at 15 m s-1. How high does it rise? (g = 9.8 m s-2)
Known: u = 15 m s-1 (up), v = 0 at max height, a = -9.8 m s-2 (taking up as positive)
Using: v2 = u2 + 2as
0 = 152 + 2(-9.8)s
s = 225 / 19.6 = 11.5 m
A cyclist travelling at 12 m s-1 brakes uniformly and stops in 30 m. Find the deceleration.
Known: u = 12 m s-1, v = 0 m s-1, s = 30 m
Using: v2 = u2 + 2as → 0 = 144 + 2a(30)
a = -144 / 60 = -2.4 m s-2 (the negative sign indicates deceleration)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Which of the following is a vector quantity?
Question 2
On a velocity-time graph, the area under the curve represents:
Question 3
A car starts from rest and accelerates at 3.0 m s-2 for 5.0 s. What is the displacement?
Question 4
Which kinematic equation does NOT contain time (t)?
Question 5
On a displacement-time graph, a straight line with a positive gradient indicates:
Key Concepts Summary
- ●Displacement is a vector quantity representing the straight-line change in position from start to finish.
- ●Velocity is the rate of change of displacement; acceleration is the rate of change of velocity.
- ●The four SUVAT equations apply only under uniform (constant) acceleration.
- ●The gradient of a s-t graph gives velocity; the gradient of a v-t graph gives acceleration.
- ●The area under a v-t graph gives displacement.