Motion

Learning goals

What students should be able to explain, calculate and investigate after completing Motion.

• Distinguish distance from displacement, and speed from velocity.
• Distinguish scalar quantities from vector quantities.
• Define acceleration and calculate it from a change in velocity.
• Draw and interpret displacement-time and velocity-time graphs.
• Use the equations of motion (SUVAT) correctly, only for constant acceleration in a straight line.
• Higher Level: Derive the constant-acceleration kinematic equations from the definitions and average-velocity relationship.
• Describe and carry out experiments to measure velocity and acceleration, including g.

Key definitions

Precise terms, symbols and SI units used in Motion.

Distance/Speed vs Displacement/Velocity

Distance and speed only tell you "how much"; displacement and velocity also tell you "which way".

If a person walks 3 m east then 3 m west, the distance travelled is 6 m, but the displacement is 0 m, because they end up back where they started. This distinction matters throughout the course: e.g. for a full lap of a circular track, the distance travelled is the circumference, but the displacement is zero.

Acceleration

Acceleration is the rate of change of velocity.

An object can accelerate by speeding up, slowing down, or changing direction at constant speed. Deceleration means acceleration opposite to the direction of velocity. Constant acceleration means the velocity changes by equal amounts in equal time intervals. Circular motion is treated in the separate Circular Motion, Satellites and Kepler’s Law topic.

SUVAT — Equations of Motion

The following five equations relate the initial velocity u, final velocity v, acceleration a, displacement s and time t.

They apply ONLY when acceleration is constant and motion is in a straight line. Nature of Science note: these equations are a mathematical model of motion. The model is only valid within its stated conditions (constant acceleration, straight line) — applying SUVAT to a situation with changing acceleration (e.g. a car easing off the accelerator gradually) would give an invalid, unsupported result.

• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• The SUVAT equations apply to motion in a straight line with constant acceleration. Choose a positive direction before substituting signed quantities.

• s = ½(u + v)t — this comes from displacement = average velocity × time, where average velocity = ½(u + v) is only valid because acceleration is constant; it is a separate SUVAT equation, not simply a rearrangement of s = ut + ½at²
• s = vt − ½at²

Conditions: the acceleration is constant and the motion is along a straight line. Here, u is initial velocity, v is final velocity, a is acceleration, s is displacement and t is time.

Two useful related forms: s = ½(u + v)t follows directly from displacement = average velocity × time. Substituting u = v − at gives s = vt − ½at².

Motion Graphs

Two graphs are used to describe motion:

A distance-time graph (using total path length) can never have a negative slope, since distance cannot decrease; a displacement-time graph can slope downward, showing motion back towards the start point. Similarly, a speed-time graph is always on or above the time axis, while a velocity-time graph can go below the axis to show motion in the negative direction.

• Distance–time graph: the slope (gradient) gives speed. A steeper slope means greater speed. A horizontal section means the object is stationary.
• Displacement–time graph: the slope (gradient) gives velocity. A horizontal section means zero velocity.
• Velocity–time graph: the slope (gradient) gives acceleration; the area between the line and the time axis gives displacement.

Vectors and Scalars

A scalar quantity has magnitude (size) only — e.g. distance, speed, mass, energy, time.

A vector has both magnitude and direction. Add vectors tip-to-tail or resolve them into perpendicular components. For example, displacements of 4 m east and 3 m north give R = √(4² + 3²) = 5 m at tan⁻¹(3/4) = 36.9° north of east. The separate Full Vectors topic develops components and two-dimensional applications.

Diagrams and formulae

Key diagrams and equations for reviewing Motion.

• Displacement-time graphs: a horizontal line means the object is at rest; a straight sloped line means constant velocity; a curved, steepening line means the object is accelerating.
• Velocity-time graph: the slope of the line gives the acceleration; the shaded area under the line gives the displacement over that time interval.

Displacement-time graphs: a horizontal line means the object is at rest; a straight sloped line means constant velocity; a curved, steepening line means the object is accelerating.

Velocity-time graph: the slope of the line gives the acceleration; the shaded area under the line gives the displacement over that time interval.

Worked Example

A step-by-step calculation applying the central relationships in Motion.

A car accelerates uniformly from rest to 20 m s⁻¹ in 8 s. Calculate (a) the acceleration, and (b) the distance travelled.

• (a) a = (v − u) ÷ t = (20 − 0) ÷ 8 = 2.5 m s⁻²
• (b) s = ut + ½at² = (0 × 8) + ½ × 2.5 × 8² = 80 m
• Check using v² = u² + 2as: 20² = 0² + 2(2.5)(s) → s = 400 ÷ 5 = 80 m ✓

Practical / Investigation

Experimental planning, measurement and evaluation for Motion.

Experiment: Investigating constant linear motion

• Use a trolley on a level runway with a ticker timer, two light gates or a motion sensor. Record position at equal time intervals, or measure speed at several positions.
• For constant motion, equal displacements occur in equal time intervals: a displacement-time graph is a straight line with constant slope, while a velocity-time graph is horizontal.
• Keep the runway level and the trolley speed low and steady; repeat measurements and account for friction or drag if the speed changes.

Experiment: Investigating the relationship between velocity and time for a trolley on a runway (constant acceleration), using a ticker timer, light gates, or a motion sensor / data logger.

• Independent variable: time. Dependent variable: velocity. Controlled variables: angle of runway/gradient, mass of trolley, surface.
• Data table: time (s) vs velocity (m s⁻¹); plot a velocity-time graph.
• Find the slope of the line to determine the acceleration.
• Sources of error: friction and air resistance are rarely fully eliminated; reaction time affects manual timing methods (light gates/data loggers remove this).
• Improvement: use light gates or a motion sensor instead of a stopwatch to reduce human reaction-time error.

Experiment: Measuring the acceleration due to gravity, g, by free fall.

• Independent variable: height fallen (or time). Dependent variable: time of fall (or height). Controlled variable: the falling object used.
• Method: release a small dense object (e.g. a steel ball) from rest and use light gates (or an electromagnet/timer circuit) to record the time to fall a measured height; repeat for several different heights.
• Since it starts from rest, s = ut + ½at² simplifies to s = ½gt² (with u = 0, a = g); plot s (y-axis) against t² (x-axis) — the slope of the straight line equals ½g, so g = 2 × slope.
• The graph should pass close to the origin; a large intercept suggests a timing or height-measurement error.
• Sources of error: air resistance (usually small for a dense ball over a short fall); reaction-time error if using a manual stopwatch instead of light gates.
• Improvement: use light gates and an electronic timer rather than a stopwatch, and release the object from an electromagnet to ensure a clean, reproducible start at u = 0.

• Trolley: obtain velocity at known times using light gates, ticker tape or a motion sensor; slope (gradient) of a v–t graph gives acceleration.
• Free fall: release from rest, measure s and electronic fall time t, plot s against t², and use g = 2 × slope (gradient).
• Repeat readings, measure distances from consistent reference points and avoid a hand stopwatch for short fall times.

Quick-Check Questions

Short retrieval questions covering the essential ideas in Motion.

1. A runner completes one lap of a 400 m circular track and returns to the start. State the distance and the displacement.

2. What does the slope of a velocity-time graph represent?

3. Why can distance-time graphs never have a negative slope?

4. Write down the three specified equations of motion.

5. State the condition under which the SUVAT equations may be applied.

6. A displacement of 6 m east and a displacement of 8 m north are added tip-to-tail. Find the magnitude of the resultant displacement.

Common mistakes and exam tips

Common misconceptions and exam-technique checks for Motion.

• Avoid: Using constant-acceleration equations when acceleration varies. Correct idea: SUVAT equations apply only when acceleration is constant over the interval.
• Avoid: Treating distance–time and displacement–time graphs as interchangeable. Correct idea: Displacement includes direction; distance does not decrease when an object reverses direction.
• Avoid: Reading one constant slope from a curved displacement–time graph. Correct idea: The instantaneous velocity is the slope of the tangent at the point.

• List u, v, a, s and t, then choose an equation containing the known quantities and the unknown.
• Choose a positive direction before using signed displacement, velocity or acceleration.
• Find displacement from the signed area under a velocity–time graph.

Exam-style questions

Five exam-style questions with worked answers on Motion.

Higher Level — Required Content

Required Higher Level content for this topic.

• Start with a = (v − u)/t to obtain v = u + at.
• For constant acceleration, average velocity = (u + v)/2, giving s = ½(u + v)t.
• Substitute v = u + at to obtain s = ut + ½at².
• Eliminate t to obtain v² = u² + 2as.

State the assumption: acceleration is constant and motion is in a straight line.

End-of-unit revision

Formulae, definitions, experiments and mixed revision questions for Motion.

• Speed = distance ÷ time; Velocity = displacement ÷ time
• a = (v − u) ÷ t
• v = u + at
• s = ut + ½at²
• v² = u² + 2as
• s = ½(u + v)t
• s = vt − ½at²
• Vector addition (tip-to-tail); for perpendicular vectors, R = √(A² + B²)

• Distance vs displacement
• Speed vs velocity
• Acceleration
• Scalar vs vector

• Displacement-time graphs for rest, constant velocity, and acceleration
• A velocity-time graph with slope = acceleration and area = displacement labelled

• Measuring velocity and acceleration using a ticker timer, light gates, or a data logger
• Determining acceleration due to gravity, g, by free fall (s vs t² graph, slope = ½g)

• Confusing s = ½(u+v)t with a rearrangement of s = ut + ½at²
• Applying SUVAT where acceleration is not constant
• Distance-time vs displacement-time graph confusion

1. A ball is thrown vertically upward at 15 m s⁻¹. Taking g = 9.8 m s⁻² (deceleration), calculate the time taken to reach its highest point.

2. Explain why the SUVAT equations cannot be applied to a car moving around a bend at constant speed.

3. A velocity-time graph shows a straight line from (0 s, 4 m s⁻¹) to (6 s, 16 m s⁻¹). Find the acceleration and the displacement over this interval.

4. State one advantage of using a motion sensor over a stopwatch when measuring the velocity of a trolley.

1. At the highest point v = 0. Using v = u + at: 0 = 15 + (−9.8)t → t = 15 ÷ 9.8 ≈ 1.53 s.

2. SUVAT requires straight-line motion; moving around a bend means the direction of velocity is continuously changing, so the motion is not in a straight line even if speed is constant.

3. a = (16 − 4) ÷ 6 = 2 m s⁻². Displacement = area under graph = ½(4+16)(6) = 60 m.

4. A motion sensor removes human reaction-time error and can record many data points automatically, giving a more precise velocity-time graph.

The table below shows which equations from this unit appear in the official State Examinations Commission Formulae and Tables book, and where.

Note: the Tables book (p. 50) gives The specification lists three equations of motion for constant acceleration. The average-velocity forms shown here are useful derived relationships, not additional required formulae.