Acceleration

1. Definition

A. Acceleration

Acceleration, a, is the rate of change of velocity with time.

a = Δ v/Δ t = v-u/t

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

B. Uniform acceleration

Uniform acceleration means acceleration is constant (velocity changes by equal amounts in equal time intervals).

2. Key Ideas

• O Level kinematics mainly uses straight-line (one-dimensional) motion.
• Average acceleration over a time interval:
  • a = Δ v/Δ t = v-u/t
• SI unit: m s⁻² (also written as m/s²).
• a = 0 means constant velocity (the object can still be moving).
• Uniform acceleration: constant a → equal change in v in equal times.
• Non-uniform acceleration: a changes with time, but you can still calculate average acceleration using Δ v/Δ t.

Scroll diagram horizontally to read all labels.

3. Detailed Explanations

A. How to calculate average acceleration (exam method)

1. Write down u, v and t (include direction/sign).
2. Use a = v-u/t.
3. Give the final answer with unit m s⁻² (and direction if asked).
4. Do a sign check: if the object is slowing down, acceleration should be opposite to the velocity direction.

Mini-example (take right as positive): u = + 4, v = + 10, t = 3.0
a = 10-4/3.0 = 2.0 m s⁻²

B. Positive/negative acceleration and “deceleration” (1D)

In 1D motion, the sign of velocity/acceleration depends on the positive direction you choose (e.g. “right is +”).

Speed decreases (“deceleration”) when velocity and acceleration have opposite signs.

C. Uniform vs non-uniform acceleration

Uniform acceleration: equal change in velocity in equal time intervals.

Example (uniform acceleration):

Non-uniform acceleration: the change in velocity per second is not constant.

Example (non-uniform acceleration):

Here, the velocity changes by + 10, + 20, -10, + 30, + 20 m s⁻¹ each second (not constant).

D. Link to velocity–time graphs

On a velocity–time graph, acceleration is the gradient:

a = Δ v/Δ t

See: Reading Kinematics Graphs (Displacement–Time & Velocity–Time).

4. Common Mistakes

A. Using speed instead of velocity

• Acceleration depends on velocity change, so direction (or sign) matters.

B. Wrong sign for Δ v

• Using u-v instead of v-u.
• Forgetting that “slowing down” can be either positive or negative acceleration depending on the direction chosen.

C. Unit and time conversion errors

• Mixing km h⁻¹ with m s⁻¹ without converting.
• Using minutes/hours without converting to seconds when using SI units.

D. “Zero acceleration means zero velocity”

• a = 0 means velocity is constant, not necessarily zero.

5. Exam Tips

A. Definition marks

• Write: “rate of change of velocity per unit time”.
• State the equation: a = Δ v/Δ t.
• Give the SI unit: m s⁻².

B. Calculation method (full marks)

• Choose a sign convention (or state directions clearly).
• Show substitution with units.
• Final answer with unit, and direction if asked.

C. Description questions

• If an object is slowing down: say “acceleration is opposite to velocity”.
• If an object is speeding up: say “acceleration is in the same direction as velocity”.

6. Worked Examples

Example 1: From rest to 20 m s⁻¹ in 10 sCore

A bus starts from rest and reaches 20 m s⁻¹ in 10 s, moving to the right. Find its average acceleration.

Show Answer

a = v-u/t; = 20-0/10; = 2.0 m s⁻²

Average acceleration = 2.0 m s⁻² to the right.

Example 2: Westwards, then stops (sign convention)Core

A car travels west at 30 m s⁻¹ and comes to rest in 5.0 s. Find its average acceleration.

Show Answer

Take east as positive, so west is negative:

• u = -30 m s⁻¹
• v = 0

a = v-u/t; = 0-(-30)/5.0; = + 6.0 m s⁻²

The positive sign means the acceleration is eastwards (opposite to the motion), so the car is slowing down.

Example 3: Moving right but slowing downCore

A car moves right at 25 m s⁻¹ but is braking and slowing down. State the direction of its acceleration.

Show Answer

The car’s velocity is to the right, but its speed is decreasing, so acceleration is opposite to the motion.

Acceleration is to the left.

Example 4: Can an object move when a = 0?Core

Can an object be moving if its acceleration is zero? Explain briefly.

Show Answer

Yes. a = 0 means velocity is constant. The object can have a constant non-zero velocity (steady motion).

Example 5: Average acceleration from dataCore

An object moves in a straight line. Its velocity changes from 2.0 m s⁻¹ to 11.0 m s⁻¹ in 6.0 s. Find its average acceleration.

Show Answer

a = v-u/t; = 11.0-2.0/6.0; = 1.5 m s⁻²

Example 6: Acceleration from a velocity–time lineCore

An object’s velocity–time graph is a straight line from (t = 0, v = 4.0) to (t = 5.0, v = 14.0), with t in seconds and v in m s⁻¹. Find its acceleration.

Show Answer

Acceleration is the gradient:

a = Δ v/Δ t; = 14.0-4.0/5.0-0; = 2.0 m s⁻²

7. Mind Stretchers

Mind stretcher 1: Direction reversal (average acceleration)Extension

Take right as positive. A trolley has velocity + 6.0 m s⁻¹ at one instant. After 4.0 s, its velocity is -2.0 m s⁻¹. Find the average acceleration.

Show Answer

a = v-u/t; = -2.0-6.0/4.0; = -2.0 m s⁻²

The negative sign means acceleration is to the left.

Mind stretcher 2: “Deceleration” does not mean negativeExtension

Take right as positive. An object moves left and slows down: u = -12 m s⁻¹, v = -4.0 m s⁻¹ in 4.0 s. Find a and state whether it is accelerating or decelerating.

Show Answer

a = v-u/t; = -4.0-(-12)/4.0; = + 2.0 m s⁻²

Acceleration is to the right (positive). Because velocity is to the left (negative), acceleration is opposite to velocity, so the object is decelerating (speed decreasing).

Mind stretcher 3: Is the acceleration uniform?Extension

An object’s velocities are:

Time / s	Velocity / m s⁻¹
0	0
1	3
2	7
3	12

Is the acceleration uniform? Explain.

Show Answer

No. The change in velocity each second is not constant:

• from 0 to 1 s: Δ v = 3
• from 1 to 2 s: Δ v = 4
• from 2 to 3 s: Δ v = 5

So acceleration increases with time (non-uniform).

8. Practice, Quiz and Next Step

Close your notes and use Acceleration in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: A cyclist's velocity changes uniformly from 4.0 m/s east to 16.0 m/s east in 6.0 s, then becomes 10.0 m/s east two seconds later.

1. Retrieve: define velocity, acceleration and sign in your own words, including units, sign or conditions where relevant.
2. Represent: Sketch a velocity–time graph for both intervals and label values, units and the change of gradient.
3. Apply: Calculate the acceleration in each interval and explain what the sign and change of magnitude mean.

Check the response before looking back

• Scalar and vector quantities are distinguished, including direction and sign.
• A graph result is inferred from its labelled axes before a formula is chosen.
• Working uses compatible units and the final statement describes the motion.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theO-Level topic checks orpractice browser for an independent re-test.