Forces - 5.6.1.5 Acceleration (GCSE Physics AQA)

Acceleration

Calculating Acceleration

Formula for Acceleration

Acceleration is the rate of change in speed of an object. We can calculate acceleration using the following formula:

Where:

• acceleration, a, in metres per second squared, m/s²
• change in velocity, ∆v, in metres per second, m/s
• time, t, in seconds, s

Question: A high-speed train accelerates at a constant rate in a straight line. The velocity of the train increases from 30 m/s to 42 m/s in 60 seconds.

1. Calculate the acceleration of the train.
Write out the formula for acceleration.

acceleration = change in velocity / time taken

2. Work out the change in velocity.

change in velocity = 42 – 30
change in velocity = 12 m/s

3. Substitute in the numbers.

acceleration = change in velocity / time taken
acceleration = 12 / 60
acceleration = 0.2 m/s²

Acceleration and Deceleration

• Acceleration means speeding up. When we say that an object is accelerating, we mean that it is speeding up. The speed is increasing and hence the velocity is changing. For example, cars accelerate when you press down on the gas pedal.
• Deceleration means slowing down. When we say that an object is decelerating, we mean that it is slowing down. This means that its speed is decreasing. Compared to acceleration, deceleration is its opposite. Sometimes, we refer to deceleration as ‘negative acceleration’.

Everyday Accelerations

• You need to know everyday speeds. Previously, we have discussed some average speeds for modes of transport including cars, trains and cycling. You will need to know these examples when answering questions about acceleration.
• You need to estimates times. When estimating acceleration, you need to know the time taken by the object as well. Since this is an estimated acceleration, you need to also estimate the time taken. We will look at this in following example.

Question: A man is planning to go on a bike ride. He travels at a normal cycle speed for most of his journey. He initially sets out from rest and it takes him a a few seconds to reach his normal cycle speed. Calculate his acceleration he has to reach the constant speed.

1. Use your own knowledge.
For this question, we are given very little information, so we need to rely heavily on our own knowledge. We know an average speed for a cyclist is about 6 m/s, and we have to estimate the time taken to reach this speed. An estimate between 5 – 20 seconds seems reasonable – we will go for 12..

Estimated average speed = 6 m/s
Time taken to reach this speed = 12 seconds

2. Write out the equation for acceleration.

acceleration = change in velocity / time taken

3. Substitute in the numbers.

acceleration = change in velocity / time taken
Acceleration = 6 / 12
Acceleration = 0.5 m/s²

Velocity-Time Graphs

Drawing Velocity-Time Graphs

Just like a distance-time graph, a velocity-time graph can also represent motion. You can see how the velocity of the object changes with time. For AQA exams, you should be able to draw and interpret these graphs, as shown below.

Question: An athlete at the start of a race. The race is along a straight track. In the first 2 seconds, the athlete accelerates constantly and reaches a speed of 9 m/s.

Complete the graph to show how the velocity of the athlete changes during the first 2 seconds of the race.

1. Pick out information.
Since this question is rather long-winded, we need to pick out the useful information.

Starting velocity = 0 m/s
Final velocity = 9 m/s
Time taken = 2 seconds

2. Draw onto the graph.
Using the information we have just found, we can now modify the diagram. We know that the time taken is 2 seconds, and that the velocity reached is 9 m/s.

Calculating Acceleration

For distance-time graphs, the gradient is the speed.

For velocity-time graphs, the gradient is the acceleration.

The question below shows how we can calculate acceleration from a velocity-time graph. If you are unsure why the gradient of the velocity-time graph tells us the acceleration, then consider the two equations below:

Question: (Following on from the previous question) Calculate the acceleration of the athlete, using the graph you have just drawn.

1. Acceleration is the gradient.
From the graph, we need to find the gradient in order to know the acceleration.

2. Find the time and velocity.
From the graph, we can work out the values for time and velocity.

Time = 2 seconds
Velocity = 9 m/s

3. Use the gradient equation.
Now that we have values for distance and time, we can use our formula change in y / change in x to find the acceleration.

gradient = change in y / change in x
gradient = 9 / 2
gradient = 4.5
gradient = acceleration
acceleration = 4.5 m/s²

Calculating Distance Travelled

We can calculate the distance travelled by an object using a velocity-time graph. Since the velocity is a vector, we can also use this graph to calculate the displacement of an object.

The area under a velocity-time graph is the distance travelled. Since speed = distance / time, the distance = speed x time (or velocity x time).

Question: The diagram shows the velocity-time graph for an object over a 10 second period.

Use the graph to calculate the distance travelled by the object in 10 seconds. Show clearly how you work out your answer.

Method 1

1. The area under the graph represents distance travelled. We know that the area under a velocity-time graph will represent the distance travelled by an object.

2. Write out the formula for area of a rectangle. The shape under the graph is a rectangle. We need to write out the formula for the area of a rectangle.

Area of a rectangle = length x width

3. Write out the lengths of the sides. Now that we have the formula, we need to work out the length and width of the rectangle.

Length = 8
Width = 6

4. Work out the area. Using these lengths, we can work out the area of the rectangle.

Area = 6 x 8
Area = 48 cm²

Method 2

1. Look at the shape. From the graph, we can see that the shape underneath is a rectangle.

2. Work out the area of the shape. Now that we have found the shape, we can work out its area. In this case, we simply multiply 8 and 6.

Sometimes it can help to split the area under the graph into multiple shapes – e.g. a square and a triangle. Work out the areas of each shape separately and then add them together.

For example, for this graph below, you can split the area under the graph into two triangles (A and C) and one square (B).

Calculating Uniform Acceleration

Formula for Uniform Acceleration

An object has uniform acceleration if the acceleration is constant throughout the journey. Therefore the speed is always changing at the same rate. Be careful – the speed is never constant, because there is always acceleration.

If there is uniform acceleration, we can calculate the acceleration using the following equation:

Where:

• final velocity, v, in metres per second, m/s
• initial velocity, u, in metres per second, m/s
• acceleration, a, in metres per second squared, m/s²
• distance, s, in metres, m

Question: A hamster in it’s ball starts from rest and accelerates to 3m/s in 6 seconds. The acceleration of the hamster is uniform.
(a) What distance has the hamster travelled?
(b) What was the acceleration?

Part A

1. Write out the appropriate formula.
In this question, we have been given a value for speed and time. Now we can use this to find a value for distance.

Speed = distance / time
Distance = speed x time

2. Substitute in the numbers.

Distance = 3 x 6
Distance = 18 metres

Part B

1. Write out the appropriate formula.
In this case, we can use the speed, distance and time. We want to find the acceleration, so the following equation is appropriate. We need to rearrange the equation.

v² − u² = 2 a s
v² − u² / 2 s= a

2. Find the numbers.
Now that we have our rearranged equation, we need to write out the values for each symbol.

s = 18 m
u = 0 m/s
v = 3 m/s
a = ?
t = 6 s

3. Substitute in the numbers.
Now that we have our equation, we can put in our numbers.

v² − u² / 2 s= a
3² – 0² / 2 x 18 = a
9 / 36 = a
a = 0.25 m/s²

Acceleration due to Gravity

• Gravity affects all objects on Earth. Gravity is a strong force that affects object on the Earth, or close to the surface of the Earth. This force stops us from floating into space.
• Gravity causes acceleration. The force of gravity causes objects to accelerate towards the surface of the Earth. When an object falls freely under gravity, it experiences an acceleration of 9.8m/s².

Terminal Velocity

If an object has reached ‘terminal velocity’, this means that it has reached a constant velocity. The forces on the object are balanced at this point.

We will now explore how a ball falling through fluid reaches terminal velocity. A fluid is referred to either a gas or liquid.

1. A ball is dropped into a fluid filled tube.
2. The weight of the ball is acting downwards, making the ball fall through the fluid very quickly.
3. As the ball accelerates, the fluid will oppose the motion. The fluid exerts a frictional force on the ball, making it more difficult for the ball to fall through the tube.
4. The frictional force gets larger and larger, making the ball accelerate less and less.
5. Eventually, the frictional force acting on the ball will become equal to the weight of the ball. At this point, the ball will not accelerate – instead, it will fall at a constant velocity.
6. The ball has now reached terminal velocity.

Velocity-Time Graphs for Terminal Velocity

1. Velocity slowly increases. On the graph, you can see that the velocity is slowly increasing as time goes on.
2. The increase in velocity decreases. As you can see from the curve, it is starting to level off. This shows us that the increase in the velocity is getting less and less.
3. Terminal velocity is reached. The graph levels off, meaning that the object is travelling at a constant velocity. This is the point at which the object has reach terminal velocity.

Forces Involved with Terminal Velocity

There are two main forces involved with terminal velocity:

• Weight – the weight of the object is a force that acts downwards due to gravity. The heavier the mass of the object, the higher the weight on earth. This force will lead to the object falling through the fluid.
• Air resistance or drag – as the object falls through the fluid, it will experience forces that oppose its motion. If the object is falling through a liquid, this force is called a drag force. If the object is falling through air, then the force is called air resistance. As the object falls faster and faster, the opposing forces will start to get larger. This will decrease the acceleration of the object, eventually leading to a constant velocity.