Kinematics: Equations of Motion

This guide covers Topic 2, Kinematics, subtopic 2.1 Equations of motion, from Cambridge International AS & A Level Physics 9702, 2025–2027 series. This is AS Level content, building directly on the scalar/vector distinction from Topic 1.

Before studying this

This resource assumes the scalar/vector distinction and vector resolution from Physical Quantities, Units and Measurement — displacement, velocity and acceleration are all vectors, and treating them as plain numbers without regard to direction is a common source of sign errors in this topic.

Syllabus coverage

CAMBRIDGE INTERNATIONAL AS & A LEVEL PHYSICS 9702 — AS Level, Topic 2

2.1 Equations of motion — defining and using distance, displacement, speed, velocity and acceleration; representing these graphically; determining displacement from the area under a velocity–time graph; determining velocity from the gradient of a displacement–time graph and acceleration from the gradient of a velocity–time graph; deriving the equations of uniformly accelerated motion in a straight line from the definitions of velocity and acceleration; solving problems with these equations, including bodies falling freely under gravity without air resistance; describing an experiment to determine the acceleration of free fall; describing motion that combines a uniform velocity in one direction with a uniform acceleration in a perpendicular direction.

Distance, displacement, speed and velocity

Distance is the total path length travelled (a scalar); displacement is the straight-line distance from start to finish, in a given direction (a vector). Speed is distance travelled per unit time (a scalar); velocity is displacement per unit time (a vector). Two objects can travel the same distance at the same speed while having very different displacements and velocities, if their paths differ.

Acceleration is the rate of change of velocity — also a vector, and note that an object can accelerate by changing direction alone, even at constant speed.

Reading motion graphs

• On a displacement–time graph, the gradient at any point gives the velocity at that instant. A horizontal section means the object is stationary; a straight sloped line means constant velocity; a curve means the velocity is changing.
• On a velocity–time graph, the gradient gives the acceleration, and the area under the graph gives the displacement over that time interval — this is signed area, so a region where velocity is negative counts as negative displacement. A horizontal line means constant velocity (zero acceleration); the area of a triangle or trapezium under a sloped line gives the displacement during that acceleration. If velocity changes sign during the interval, total distance travelled requires adding the magnitudes of the areas above and below the time axis separately, rather than reading off a single signed area.

Being asked to determine one quantity from the “area under” or “gradient of” a graph is one of the most frequently tested skills in this subtopic — practise reading both types of graph before touching the equations below.

The equations of motion

For uniformly accelerated motion in a straight line, with initial velocity u, final velocity v, acceleration a, displacement s and time t:

v = u + at
s = ut + ½at²
v² = u² + 2as
s = ½(u + v)t

These are derived directly from the definitions of velocity (rate of change of displacement) and acceleration (rate of change of velocity) — you should be able to derive the first from the definition of acceleration, and the others by combining it with the definition of velocity, since “derive” is an explicit syllabus command word here, not just “recall.”

Worked example. A ball is dropped from rest and falls freely (ignore air resistance) for 2.0 s. Using g = 9.81 m s⁻² and s = ut + ½at² with u = 0:

s = 0 + ½ × 9.81 × 2.0² = 19.6 m

Its velocity at that point, from v = u + at: v = 0 + 9.81 × 2.0 = 19.6 m s⁻¹.

Determining g experimentally

A standard method drops an object through a known, measured height and times the fall electronically (a light gate or timer released by an electromagnet), then uses s = ½gt² (since u = 0) rearranged to g = 2s/t². Repeating the drop and averaging t reduces the effect of random timing error.

Motion with a uniform velocity and a perpendicular uniform acceleration

Where a uniform velocity in one direction and a uniform acceleration in a perpendicular direction act at the same time — for example, a ball thrown horizontally that also falls under gravity — the two motions are independent and can be analysed separately, using the equations above in each direction, then combined only if a resultant displacement or velocity is asked for. This is the same resolving idea from Topic 1, applied to motion rather than to a single force.

Common mistakes

• Mixing up “distance” and “displacement,” or “speed” and “velocity,” when a question explicitly distinguishes them — a returning journey has non-zero distance but can have zero displacement.
• Reading a velocity–time graph’s gradient when displacement is asked for (or vice versa) — gradient gives velocity or acceleration; area gives displacement.
• Forgetting that free fall without air resistance means constant acceleration, so all four equations of motion apply directly — with air resistance present, they don’t (see the next topic, Dynamics, for terminal velocity).
• Treating the horizontal and vertical components of motion as dependent on each other, when the syllabus explicitly treats them as independent directions that are only combined at the end if needed.

Quick revision checklist

• Distance vs. displacement, speed vs. velocity (scalar/vector pairs)
• Reading gradient and area from displacement–time and velocity–time graphs
• The four equations of uniformly accelerated motion, and deriving the first from the definitions of velocity and acceleration
• Applying the equations to free fall (u = 0, a = g)
• An experimental method to determine g
• Treating perpendicular uniform-velocity and uniform-acceleration motions independently

Related resources

• Physical Quantities, Units and Measurement — the vector resolution this topic builds on
• Dynamics: Newton’s Laws and Momentum — the next AS topic, connecting motion to force
• Cambridge AS & A Level Physics hub

Written against Cambridge International AS & A Level Physics 9702, 2025–2027 series. Always check the current syllabus for your examination year.