Kinematics of Motion in a Straight Line

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities (restricted to motion in one dimension only)

  2. Sketch and interpret displacement–time graphs and velocity–time graphs, and in particular appreciate that:

    • the area under a velocity–time graph represents displacement
    • the gradient of a displacement–time graph represents velocity
    • the gradient of a velocity–time graph represents acceleration

  3. Use differentiation and integration with respect to time to solve simple problems concerning displacement, velocity and acceleration

  4. Use appropriate formulae for motion with constant acceleration in a straight line (including problems with more than one equation and multiple particles)

1. Scalar and Vector Quantities

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What are Scalars and Vectors?

Scalar quantities have only a magnitude (size). Think of them as just numbers with units.

Vector quantities have both magnitude and direction. They tell you not just "how much" but also "which way".

Distance vs Displacement

Distance is a scalar quantity. It measures the total length of the path travelled, regardless of direction. Think of it like the reading on a car's odometer – it just keeps adding up.

Example: If you walk 50 m forward, then turn around and walk 30 m back, the total distance you've travelled is 50 + 30 = 80 m.

Displacement is a vector quantity. It measures how far you are from your starting point, taking direction into account. We use a fixed reference point called the origin to measure from.

Example: Using the same journey above, if we define "forward" as positive, your displacement is +50 m – 30 m = +20 m. You're 20 m ahead of where you started.

Key point: You can travel a long distance but have zero displacement if you end up back where you started (like the Duke of York's men in the old rhyme who marched up the hill and back down again!).

Speed vs Velocity

Speed is a scalar quantity. It tells you how fast something is moving, but not which direction.

speed=distance travelledtime taken\text{speed} = \frac{\text{distance travelled}}{\text{time taken}}

When speed is constant, we use this formula directly. When speed changes, we calculate:

average speed=total distancetotal time\text{average speed} = \frac{\text{total distance}}{\text{total time}}

Units: metres per second (m s⁻¹) or sometimes kilometres per hour (km h⁻¹)

Velocity is a vector quantity. It tells you how fast the displacement is changing and includes direction.

velocity=change in displacementtime taken\text{velocity} = \frac{\text{change in displacement}}{\text{time taken}}

When velocity is constant, use this formula directly. When velocity changes:

average velocity=net displacementtotal time\text{average velocity} = \frac{\text{net displacement}}{\text{total time}}

Units: metres per second (m s⁻¹)

Example: A car drives 9 km in 15 minutes at constant speed.

  • Convert to SI units: 9 km = 9000 m, 15 minutes = 900 s
  • Using s = vt: 9000 = v × 900
  • Speed: v = 10 m s⁻¹

Important: In one-dimensional motion, we choose one direction as positive. Movement in the opposite direction is negative.

Example: A cyclist travels at 5 m s⁻¹ for 30 s (displacement = 5 × 30 = 150 m), then turns around and travels at 3 m s⁻¹ for 10 s in the opposite direction (displacement = –3 × 10 = –30 m). Total displacement = 150 + (–30) = 120 m in the original direction.

Acceleration

Acceleration is a vector quantity. It measures how quickly velocity is changing.

a=v−uta = \frac{v - u}{t}

where:

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

Units: metres per second per second, written as m s⁻²

Positive acceleration means velocity is increasing in the positive direction (speeding up if moving forward, or slowing down if moving backward).

Negative acceleration means velocity is decreasing in the positive direction (slowing down if moving forward, or speeding up if moving backward).

Deceleration is a term sometimes used to mean "slowing down" – it's a negative acceleration when moving in the positive direction.

Example: A parachutist falls from rest (u = 0) to 49 m s⁻¹ in 5 s.

  • Acceleration: a = (49 – 0) ÷ 5 = 9.8 m s⁻²

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