O-Level Pure Physics: Kinematics — Distance-Time & Velocity-Time Graphs and Equations of Motion

Why Kinematics Matters in O-Level Pure Physics

Kinematics — the study of motion — appears in Section A (short-answer) and Section B (structured questions) of the O-Level Pure Physics paper almost every year. A well-prepared student can pick up full marks on kinematics reliably; an underprepared student loses marks not from lacking physics knowledge but from misreading graphs or substituting into the wrong equation.

This guide gives you a systematic method for each type of kinematics question: graph interpretation, graph-to-value extraction, and equation selection. Master these three skills and kinematics becomes one of your highest-scoring topics.

Scalar vs Vector: Getting the Basics Right

Before graphs and equations, clarify the four fundamental quantities:

• Distance — total path length, always positive (scalar).
• Displacement — change in position with direction; can be negative (vector).
• Speed — rate of change of distance, always positive (scalar).
• Velocity — rate of change of displacement with direction; can be negative (vector).

Questions that ask for "distance" and "displacement" separately want different numerical answers. Never substitute them interchangeably into equations of motion.

Part 1: Distance-Time (d-t) Graphs

What the gradient tells you

On a distance-time graph, the gradient at any point equals the speed of the object at that point.

• Flat horizontal line — gradient = 0, so speed = 0. The object is stationary.
• Straight line with positive gradient — constant speed in the forward direction.
• Straight line with negative gradient — constant speed in the reverse direction (the object is returning towards the starting point).
• Curved line, gradient increasing — the object is accelerating (getting faster).
• Curved line, gradient decreasing — the object is decelerating (slowing down).

Reading speed from a d-t graph

For a straight-line segment: draw a right-angled triangle under the line and calculate:

Speed = (change in distance) ÷ (change in time)

For a curved section, the instantaneous speed at a point equals the gradient of the tangent drawn at that point. In O-Level questions this is typically given or you are asked to draw the tangent and estimate the gradient.

Common d-t graph scenarios

1. Object moves away, stops, then returns → positive slope, flat segment, negative slope.
2. Object accelerates then travels at constant speed → concave-up curve, then straight line.
3. Two objects whose lines intersect — the intersection is the point where they meet.

Part 2: Velocity-Time (v-t) Graphs

Two things every v-t graph tells you simultaneously

• Gradient of any segment = acceleration at that point.
• Area under the graph = displacement for that time interval.

Reading acceleration

For a straight-line segment:

Acceleration = (change in velocity) ÷ (change in time)

Units: m s⁻². A negative value means deceleration (assuming motion in the positive direction). A line sloping upward = acceleration; sloping downward = deceleration.

Reading displacement from the area

Break the v-t graph into recognisable shapes:

• Rectangle (constant velocity, v × t).
• Triangle (uniform acceleration from or to zero, ½ × t × v).
• Trapezium (uniform acceleration between two non-zero velocities, ½(u + v) × t).

If the graph dips below the time axis, the object moves in the opposite direction. Area below the axis counts as negative displacement — it reduces total displacement but adds to total distance travelled.

Key v-t graph shapes to recognise instantly

• Horizontal line at v > 0: constant velocity, zero acceleration.
• Straight line sloping up from origin: uniform acceleration from rest.
• Straight line sloping down to the axis: uniform deceleration to rest.
• Line crossing the time axis: object reverses direction.

Part 3: The Four Equations of Motion

These equations apply only when acceleration is constant (uniform). The five variables are:

• u — initial velocity (m s⁻¹)
• v — final velocity (m s⁻¹)
• a — acceleration (m s⁻²)
• s — displacement (m)
• t — time (s)

The four equations, each omitting one variable:

1. v = u + at  (no s)
2. s = ut + ½at²  (no v)
3. v² = u² + 2as  (no t)
4. s = ½(u + v)t  (no a)

Choosing the right equation: a 3-step method

1. List the known variables from the question (always at least three).
2. Identify the unknown you need to find.
3. Pick the equation that contains both the unknown and all known variables, and does not require a fourth unknown you have not been given.

Example: given u, a, t — asked for v. Use v = u + at (equation 1, contains u, a, t, v but not s). Given u, v, s — asked for a. Use v² = u² + 2as (equation 3, contains u, v, a, s but not t).

Sign convention for equations of motion

Choose a positive direction at the start of each problem and state it explicitly. Upward is commonly taken as positive for projectile problems. If the acceleration due to gravity acts downward, write a = −10 m s⁻² (or +10 if you take downward as positive). Inconsistent signs are the single biggest source of errors in multi-step kinematics questions.

Worked Examples

Example 1 — v-t Graph: Finding Displacement

A car accelerates uniformly from rest to 20 m s⁻¹ in 10 s, maintains this speed for 30 s, then decelerates uniformly to rest in 5 s. Find (a) the acceleration during the first phase and (b) the total distance travelled.

(a) a = (v − u) ÷ t = (20 − 0) ÷ 10 = 2 m s⁻²

(b) Total distance = sum of three areas:

• Triangle (0–10 s): ½ × 10 × 20 = 100 m
• Rectangle (10–40 s): 30 × 20 = 600 m
• Triangle (40–45 s): ½ × 5 × 20 = 50 m

Total = 100 + 600 + 50 = 750 m

Example 2 — Equations of Motion: Free Fall

A ball is dropped from rest from a height of 80 m. Taking g = 10 m s⁻² and ignoring air resistance, find (a) the time to reach the ground and (b) the speed at impact.

Known: u = 0, a = 10 m s⁻², s = 80 m. For (a), unknown is t — no v, so use s = ut + ½at².

80 = 0 + ½ × 10 × t² → t² = 16 → t = 4 s

For (b): use v² = u² + 2as = 0 + 2 × 10 × 80 = 1600 → v = 40 m s⁻¹

Example 3 — Distance vs Displacement

A runner completes one full lap of a 400 m track in 80 s. Find (a) distance, (b) displacement, (c) average speed, (d) average velocity.

• (a) Distance = 400 m
• (b) Displacement = 0 m (start and end at same point)
• (c) Average speed = 400 ÷ 80 = 5 m s⁻¹
• (d) Average velocity = 0 ÷ 80 = 0 m s⁻¹

Exam Traps to Avoid

Trap 1: Confusing distance with displacement

Distance is always positive and equals total path length. Displacement is a vector and is zero if the object returns to its start. Never use them interchangeably — examiners award separate marks for each.

Trap 2: Reading the wrong gradient on a v-t graph

Calculate the gradient over the specific segment asked about, not across the entire graph. Identify the correct time interval before drawing your triangle.

Trap 3: Incorrect sign for acceleration due to gravity

Clearly state your sign convention. If upward is positive, then a = −10 m s⁻². Substituting +10 when you have defined upward as positive turns every subsequent answer negative for the wrong reason.

Trap 4: Forgetting the area below the time axis on a v-t graph

When the graph dips below the axis, that area represents motion in the reverse direction. Add it to the total distance but subtract it from the net displacement. Treating all areas as positive gives displacement = distance — which is only true if the object never reverses direction.

Trap 5: Using equations of motion for non-uniform acceleration

The four equations require constant acceleration. A curved v-t graph indicates changing acceleration; in that case, treat the question as a graph-reading problem, not an equation-substitution problem.

How to Practise Kinematics Effectively

1. Sketch the motion first. Even when the question gives only numbers, draw a rough v-t graph before selecting an equation. Visualising the motion prevents sign errors.
2. Label all five variables before choosing an equation. Write u, v, a, s, t in a column, fill in what is given, mark the unknown, then select.
3. Drill area calculations on non-standard shapes. MOE setters combine rectangles and triangles creatively. Practice decomposing any v-t graph until it is automatic.
4. Work through at least five past-year O-Level Physics kinematics questions. Focus on multi-part structured questions that chain graph reading with equation use.

With systematic graph reading and a reliable equation-selection routine, kinematics can become one of the most mark-efficient topics in your O-Level Pure Physics paper.