Acceleration

4. Deceleration

What is deceleration?

Deceleration occurs when an object’s speed decreases over time. In everyday language, the object is slowing down.

Examples include:

• A car slowing as its driver applies the brakes.
• A bicycle slowing as it travels uphill.
• A ball slowing as it rises after being thrown vertically upwards.
• A train reducing its speed before reaching a station.

Deceleration is a type of acceleration. In physics, acceleration means a change in velocity over time, and that change can involve an increase in speed, a decrease in speed, or a change in direction.

An object does not need to stop completely to decelerate. A car slowing from 25 m/s to 15 m/s is decelerating throughout that change.

Speed, velocity and acceleration

To understand deceleration clearly, distinguish between three quantities:

• Speed describes how quickly an object moves.
• Velocity describes its speed and direction.
• Acceleration describes how quickly its velocity changes.

Speed is a scalar quantity and cannot be negative. Velocity and acceleration are vector quantities, so their signs can indicate direction along a chosen axis.

For motion along a straight line:

An object slows down when its acceleration acts in the opposite direction to its velocity.

For example, a car travelling east decelerates if its acceleration is directed west.

The car can still be moving east while accelerating west. Acceleration describes how the velocity is changing, not necessarily the direction in which the object is moving.

Calculating acceleration during deceleration

Average acceleration is calculated using:

a = (v − u) / Δt

Where:

• a = average acceleration, in metres per second squared, m/s².
• u = initial velocity, in metres per second, m/s.
• v = final velocity, in metres per second, m/s.
• Δt = time taken, in seconds, s.

For constant acceleration, this average is also the acceleration throughout the interval.

If we choose the initial direction of motion as positive, an object slowing down without reversing direction has v < u. Its calculated acceleration is therefore negative.

Worked example: a car slowing down

A car slows from 20 m/s to 8 m/s in 4 s. Calculate its average acceleration, taking its direction of travel as positive.

a = (v − u) / Δt
a = (8 − 20) / 4
a = −12 / 4
a = −3 m/s²

The negative sign means the acceleration acts opposite to the chosen positive direction.

The magnitude of the deceleration is 3 m/s².

If the deceleration is constant, the car’s speed decreases by 3 m/s each second.

Does negative acceleration always mean deceleration?

No. Negative acceleration means acceleration in the negative direction. Whether an object speeds up or slows down depends on the direction of its velocity as well.

The rule is:

• Velocity and acceleration in the same direction: speeding up.
• Velocity and acceleration in opposite directions: slowing down.

Worked example: slowing down with positive acceleration

A trolley moves west. Taking east as positive, its velocity changes from −10 m/s to −4 m/s in 3 s.

a = [−4 − (−10)] / 3
a = 6 / 3
a = +2 m/s²

Its speed decreases from 10 m/s to 4 m/s, so it is decelerating.

However, its acceleration is positive because the acceleration acts east while the trolley moves west.

Worked example: speeding up with negative acceleration

A trolley’s velocity changes from −4 m/s to −10 m/s in 3 s.

a = [−10 − (−4)] / 3
a = −6 / 3
a = −2 m/s²

Its speed increases from 4 m/s to 10 m/s. It is speeding up, despite having negative acceleration.

Uniform and non-uniform deceleration

Uniform deceleration means speed decreases by equal amounts in equal time intervals.

For example:

The speed decreases by 4 m/s every second. The magnitude of the deceleration is therefore 4 m/s².

For straight-line motion with the direction of travel taken as positive, the acceleration is −4 m/s².

Non-uniform deceleration means the rate of slowing changes. A driver may brake gently at first and then more strongly.

In that case, the equation a = (v − u) / Δt gives the average acceleration over the selected interval. It does not necessarily give the acceleration at every moment.

Deceleration on a speed–time graph

On a speed–time graph:

• A downward-sloping line shows decreasing speed.
• A straight downward-sloping line shows uniform deceleration.
• A curved line that falls shows non-uniform deceleration.
• A horizontal line shows constant speed.

The steeper the downward slope, the greater the rate of decrease in speed.

For the table above, plot time horizontally and speed vertically. The points lie on a straight line from (0, 16) to (4, 0).

Its gradient is:

Gradient = change in speed / change in time
Gradient = (0 − 16) / (4 − 0)
Gradient = −4 m/s²

This tells us that speed decreases at 4 m/s each second.

A speed–time graph cannot extend below the time axis because speed cannot be negative.

Deceleration on a velocity–time graph

The gradient of a velocity–time graph gives acceleration:

Acceleration = change in velocity / change in time

To identify deceleration, look for velocity moving towards zero.

• Above the time axis, a line sloping down towards zero shows slowing down.
• Below the time axis, a line sloping up towards zero shows slowing down.

This is why a downward slope does not always mean deceleration. Below the axis, a line becoming more negative represents increasing speed in the negative direction.

Example: slowing down and then reversing

Suppose a trolley has an initial velocity of +6 m/s and a constant acceleration of −2 m/s².

During the first 3 seconds, the trolley slows down.

At 3 seconds, it is momentarily stationary.

After 3 seconds, it moves in the opposite direction and speeds up.

Its acceleration remains −2 m/s² throughout. The same acceleration first causes deceleration and then causes an increase in speed.

Finding the time taken to stop

When an object stops, its final velocity is zero.

Rearrange the acceleration equation:

Δt = (v − u) / a

Worked example

A cyclist travels at 12 m/s and slows uniformly with an acceleration of −3 m/s². How long does it take to stop?

Δt = (0 − 12) / (−3)
Δt = 4 s

The stopping time is positive because both the numerator and denominator are negative.

Always keep a consistent sign convention throughout the calculation.

Calculating distance travelled while slowing down

An object continues travelling while it decelerates.

For straight-line motion with constant acceleration and no change of direction, average speed is:

Average speed = (initial speed + final speed) / 2

Then:

Distance = average speed × time

Worked example

A car slows uniformly from 20 m/s to rest in 5 s. How far does it travel while braking?

Average speed = (20 + 0) / 2
Average speed = 10 m/s

Distance = 10 × 5
Distance = 50 m

The same result comes from the triangular area under its speed–time graph:

Distance = ½ × base × height
Distance = ½ × 5 × 20
Distance = 50 m

The simple average of the initial and final speeds is appropriate here because the deceleration is constant. It should not be assumed for every slowing-down motion.

What causes deceleration?

A moving object decelerates when its resultant force has a component opposite to its motion.

For straight-line motion:

F = ma

The resultant force and acceleration point in the same direction.

Examples include:

• Braking: Frictional forces help slow a vehicle.
• Air resistance: Drag can slow an object moving through air.
• Water resistance: Drag slows a swimmer who stops pushing through the water.
• Gravity: A component of weight slows an object moving uphill, or gravity slows a ball moving vertically upwards.

A force can oppose motion without causing deceleration if other forces balance it. For example, a car travelling at constant speed may have a driving force that balances resistive forces.

It is the resultant force, rather than any single force considered alone, that determines the acceleration.

Real-world connection: a ball thrown upwards

Ignoring air resistance, a ball near Earth’s surface has an approximately constant downward acceleration of 9.8 m/s².

If upwards is positive:

a ≈ −9.8 m/s²

As the ball rises:

• Its velocity is upwards.
• Its acceleration is downwards.
• Its speed decreases.

At the highest point:

• Its instantaneous velocity is zero.
• Its acceleration is still downwards.

As it falls:

• Its velocity and acceleration are both downwards.
• Its speed increases.

Zero velocity does not necessarily mean zero acceleration.

Real-world connection: braking and stopping

A vehicle’s total stopping distance includes:

• Thinking distance: Distance travelled before braking begins.
• Braking distance: Distance travelled while the brakes slow the vehicle.

Under a simplified constant-deceleration model, a higher initial speed increases both the stopping time and braking distance.

Using the equation:

v² = u² + 2as

and setting v = 0 gives:

s = u² / (2d)

Here, d is the positive magnitude of the deceleration.

For the same deceleration, doubling the initial speed quadruples the braking distance. This is because braking distance depends on the square of the initial speed.

Actual braking also depends on conditions such as tyre grip, road surface and braking performance.

Common misconceptions

• “Deceleration means moving backwards.” It means decreasing speed, regardless of direction.
• “Negative acceleration always means slowing down.” Compare the signs of velocity and acceleration.
• “A stopped object cannot be accelerating.” An object can have zero velocity at an instant while its velocity is changing.
• “Deceleration must be constant.” The rate of slowing can vary.
• “A force opposite to motion always causes slowing.” Other forces may balance it; consider the resultant force.
• “An object travels no distance while stopping.” It continues moving throughout the slowing-down process.

Did you know?

A car’s acceleration can point towards the rear of the car while the car continues moving forwards.

This happens during braking. The backward acceleration reduces the forward velocity rather than immediately reversing the car’s motion.

Key terms

• Deceleration: A decrease in speed over time.
• Speed: The rate at which distance is travelled.
• Velocity: Speed in a specified direction.
• Acceleration: The rate of change of velocity.
• Average acceleration: Total change in velocity divided by the time taken.
• Uniform deceleration: A constant rate of decrease in speed.
• Resultant force: The vector sum of all forces acting on an object.
• Stopping time: The time taken for a moving object to reach rest.
• Braking distance: The distance travelled while a vehicle slows under braking.
• Sign convention: A chosen system for representing opposite directions with positive and negative values.

Key takeaways

• Deceleration means decreasing speed.
• An object slows down when velocity and acceleration act in opposite directions.
• Calculate average acceleration using a = (v − u) / Δt.
• Negative acceleration does not always mean deceleration.
• On a speed–time graph, deceleration appears as a downward slope.
• On a velocity–time graph, deceleration occurs when velocity moves towards zero.
• An object can be momentarily stationary while still accelerating.
• Deceleration is determined by the resultant force and the direction of motion.