# Acceleration

## Calculating Acceleration

Acceleration is a measure of the change in an object’s velocity over time. In other words, how quickly the object is speeding up or slowing down.

Acceleration is a vector quantity, therefore, we should provide both magnitude and direction.

As with velocity and force, the direction can be indicated by positive or negative numbers.

Acceleration = Change in velocity / time.

Or _a = v -u / t _

where a is acceleration,__ v__ is the final velocity, u is the starting velocity and t is time.

Acceleration is measured in meters per second per second m/s/s or ms-2

Example: An object is accelerating at 10ms-2 from rest. This means that for each second it is accelerating it will add 10 ms-1 to its velocity.

Starts at 0 m/s

After 1 second = 10m/s

After 2 seconds = 20 m/s

After 3 seconds = 30 m/s

Worked Example 1

A car is travelling at 15 m/s, it accelerated to 25 m/s over a period of 4 seconds. What is its acceleration?

a = v - u /t

_a = _25 - 15 / 4

a = 2.5 m/s/s

Worked Example 2

A plane slows down on landing from 45 m/s to 3 m/s in 12 seconds, what is its acceleration.

a = v - u /t

_a _= 3 - 45 / 12

_a _= -42 / 12

_a = -3.5 m/s/s _The negative result indicates the object is decelerating.

The plane reduces its velocity by 3.5 m/s for each second it is braking.

How far did the plane travel during this deceleration on landing?

To calculate this information we need to use an additional equation.

v2 - u2 = 2a 𝑥 where 𝑥 is the distance travelled.

We can rearrange this to give:

𝑥 = v2 - u2 / 2 a

𝑥 = (3)2 - (45)2 / (2 x -3.5)

𝑥 = -2016 / -7

𝑥 = 288 m

## Velocity Time Graphs

Velocity-time graphs are the second way to represent the motion of an object using a graph.

Look at the example above. What is a happening during each section of the object’s journey?

A: The value of the velocity is 0 m/s so the object is stationary.

A - B: The velocity is increasing at a constant rate, this is constant positive acceleration.

B - C: The velocity remains at 20 m/s, this is constant velocity.

C - D: The velocity is decreasing at a constant rate, this is constant negative acceleration.

D: The velocity is briefly 0 m/s, so for a fraction of a second, the object has stopped.

D - E: The velocity is now negative and increasing in value. It is accelerating in the opposite direction to its previous movement.

E - F: A constant negative velocity. The object is travelling in the opposite direction at a constant velocity.

G onwards it is reducing its velocity to 0 m/s so it is decelerating.

Notice that on a velocity-time graph the direction of movement is represented by the positive and negative scales of the 𝑦-axis.

The gradient of the line represents the acceleration of the object, a change in velocity over time.

Example: Acceleration from A to B. Change in velocity 20 m/s in 5 seconds, so the acceleration is 20 ÷ 5 = 4 m/s/s.

If the line on a velocity-time graph is curved it indicates a changing rate of acceleration. In this example, the acceleration is greatest at the start of the motion and gradually decreases from about 5 seconds.

This would be the shape of a velocity-time graph for an object falling.

Calculating the Distance Travelled from a Velocity-Time Graph

The distance travelled is equal to the area between the plotted line and the time axis.

The graph above has been divided into 3 sections. Two right angle triangles and a rectangle.Using the appropriate formula we can calculate the area of each shape, then add them together to find the area under the line. This is the distance travelled.

The area of a right angle triangle = ½ height x width.

So the three areas, and the distance travelled are:

½ (20 x 5) + (5x 20) + ½ (2 x 20) = 170 m

You will not be asked to calculate this for a graph with a curved line.

## Examples of Acceleration

When an object falls it is pulled towards the earth by the force of gravity because the force is constant and unbalanced, the object will accelerate towards the earth. This is known as free-fall due to gravity.

The pull of gravity on earth is constant this means that the acceleration is constant too.

__Acceleration due to gravity is 10 m/s/s __

and has the symbol g.

Example - A parachutist jumps from a plane and in the first 5 seconds they accelerate to 50 m/s, (that is nearly 112 mph!)

Examples of typical rates of acceleration:

A large passenger jet on takeoff: 1.8 m/s/s

A small passenger jet on takeoff: 2.5 m/s/s

An average family car: 3 to 4 m/s/s

A person starting to walk 2.5 to 3 m/s/s

A ball is dropped and falls for 4.3 seconds, assuming no air resistance, what was its velocity at the end of the fall?