Equations of motion

AS Level Physics · Kinematics. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Displacement is distance in a stated direction from a starting point. Velocity is the rate of change of displacement and acceleration is the rate of change of velocity. All three are vectors, so choose a positive direction and give opposite directions a minus sign.

Graphs carry the same information. The gradient of a displacement–time graph is velocity. The gradient of a velocity–time graph is acceleration and the area under it is displacement.

The four equations of motion work only for constant acceleration in a straight line. Free fall without air resistance has constant acceleration g = 9.81 m s−2 downwards. For a projectile, treat horizontal and vertical motion separately: horizontal velocity stays constant, vertical motion has acceleration g.

Rules to remember

Common mistake

Forgetting signs when an object moves up and then down. If up is positive, then g is −9.81 m s−2 for the whole flight, including at the highest point.

Worked example

A ball is kicked from level ground at 20 m s−1 at 30° above the horizontal. Ignoring air resistance, how far away does it land? (sin 30° = 0.500, cos 30° = 0.866, g = 9.81 m s−2)

  1. Vertical component = 20 × 0.500 = 10 m s−1; horizontal component = 20 × 0.866 = 17.3 m s−1.
  2. Vertically, displacement is zero on landing: 0 = 10t − ½ × 9.81 × t2, so t = (2 × 10) ÷ 9.81 = 2.04 s.
  3. Horizontal distance = horizontal velocity × time = 17.3 × 2.04 = 35.3 m.

Answer: About 35 m.

Practice questions

Try each one, then open the answer.

1. On a velocity–time graph, the area between the line and the time axis represents

  1. A
    acceleration
  2. B
    displacement
  3. C
    speed
  4. D
    force
Show answer

Answer: B. Area = velocity × time = displacement. The gradient gives acceleration.

2. A car accelerates uniformly from 5.0 m/s to 15 m/s over a distance of 100 m. What is its acceleration?

  1. A
    0.10 m/s2
  2. B
    0.50 m/s2
  3. C
    1.0 m/s2
  4. D
    2.0 m/s2
Show answer

Answer: C. v2 = u2 + 2as: 225 = 25 + 200a, so a = 1.0 m/s2.

3. A velocity–time graph is a straight line with negative gradient crossing the time axis. What does the crossing point represent?

  1. A
    the object stops permanently
  2. B
    the object momentarily stops and reverses direction
  3. C
    the object has zero acceleration
  4. D
    the displacement is zero
Show answer

Answer: B. Velocity changes sign at the axis: the object is instantaneously at rest and then moves the other way. The acceleration (gradient) is constant and non-zero.

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