🚀 Acceleration
The object speeds up. The final velocity is bigger than the initial velocity, so v − u is positive and the acceleration is a positive number.
Movement and Position » Acceleration
What you'll learn this session
Study time: 30 minutes
Edexcel spec: 1.6
- What acceleration and deceleration mean
- The equation a = (v − u) / t, which you must learn
- Why acceleration is measured in m/s2
- How to rearrange the equation to find v, u or t
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Unlock This CourseWhat is acceleration?
A motorbike pulls away from traffic lights. One second it is crawling, a few seconds later it is flying along. Its velocity is changing, and physics has a word for how quickly that happens: acceleration.
Acceleration is the change in velocity every second. A large acceleration means the velocity changes quickly. A small acceleration means it changes slowly. If the velocity is not changing at all, the acceleration is zero, however fast the object is moving. A jet cruising at a steady 250 m/s in a straight line has no acceleration.
Key terms:
- Velocity: speed in a given direction. It is measured in m/s, just like speed.
- Acceleration: the change in velocity divided by the time taken for the change.
- Deceleration: slowing down. The velocity gets smaller, so the acceleration is negative.
- Initial velocity (u): the velocity at the start of the change.
- Final velocity (v): the velocity at the end of the change.
- Metre per second squared (m/s2): the unit of acceleration.
The equation you must learn
This equation is not given in the exam, so you have to know it by heart, in words and in symbols.
Learn this
acceleration = change in velocity / time taken
a = (v − u) / t
| Symbol | Quantity | Unit |
|---|---|---|
| a | acceleration | metre per second squared (m/s2) |
| v | final velocity | metre per second (m/s) |
| u | initial velocity | metre per second (m/s) |
| t | time taken | second (s) |
The change in velocity is always final velocity minus initial velocity, which is v − u. Work out the bracket first, then divide by the time.
Two phrases in exam questions give you a velocity without stating a number. Starts from rest means u = 0. Comes to rest or stops means v = 0.
Worked example 1: finding acceleration
A motorbike speeds up from 4 m/s to 22 m/s in 6 s. Calculate its acceleration.
a = (v − u) / t
a = (22 − 4) / 6
a = 18 / 6
a = 3 m/s2
What does m/s2 mean?
Acceleration is a change in velocity (in m/s) divided by a time (in s). That gives metres per second, per second, written m/s2 and said as "metres per second squared".
So the motorbike's acceleration of 3 m/s2 means its velocity goes up by 3 m/s every second. This table shows the pattern.
| Time in s | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|
| Velocity in m/s | 4 | 7 | 10 | 13 | 16 | 19 | 22 |
Before you use the equation, check the units. Velocities must be in m/s and the time in s. If a question gives a time in minutes, change it to seconds first.
Speeding up and slowing down
🛑 Deceleration
The object slows down. The final velocity is smaller than the initial velocity, so v − u is negative and the acceleration is a negative number.
Worked example 2: a deceleration
A lorry travelling at 25 m/s brakes and slows to 10 m/s in 5 s. Calculate its acceleration.
a = (v − u) / t
a = (10 − 25) / 5
a = −15 / 5
a = −3 m/s2
The minus sign shows the lorry is slowing down. You can write "an acceleration of −3 m/s2" or "a deceleration of 3 m/s2". Both mean the same thing: the lorry loses 3 m/s of velocity every second.
Rearranging the equation
Exam questions do not always ask for the acceleration. You may be given a and asked for the final velocity, the initial velocity or the time. Rearrange in small steps, doing the same thing to both sides.
Start with a = (v − u) / t and multiply both sides by t. This gives v − u = a × t. In words, change in velocity = acceleration × time taken. From there you can reach all three versions.
| To find | Use | In words |
|---|---|---|
| final velocity | v = u + (a × t) | initial velocity plus the change in velocity |
| initial velocity | u = v − (a × t) | final velocity minus the change in velocity |
| time taken | t = (v − u) / a | change in velocity divided by acceleration |
You only need to memorise a = (v − u) / t. In the exam, write that equation first, then rearrange it.
Worked example 3: finding the final velocity
An aeroplane starts from rest at the end of a runway. It accelerates at 2.5 m/s2 for 30 s. Calculate its final velocity.
"Starts from rest" means u = 0.
v = u + (a × t)
v = 0 + (2.5 × 30)
v = 75 m/s
Worked example 4: finding the time taken
A horse accelerates from 3 m/s to 12 m/s. Its acceleration is 1.5 m/s2. Calculate the time taken.
t = (v − u) / a
t = (12 − 3) / 1.5
t = 9 / 1.5
t = 6 s
Worked example 5: finding the initial velocity
A speedboat accelerates at 2 m/s2 for 4 s and reaches a velocity of 14 m/s. Calculate its initial velocity.
u = v − (a × t)
u = 14 − (2 × 4)
u = 14 − 8
u = 6 m/s
Always check that your answer makes sense. The speedboat was speeding up, so its initial velocity must be smaller than 14 m/s, and 6 m/s is.
Common mistakes
- Subtracting the wrong way round. It is always v − u, final minus initial. Writing u − v gives the wrong sign.
- Dividing a velocity by the time. Acceleration uses the change in velocity. A scooter going from 2 m/s to 8 m/s in 3 s has an acceleration of 6 / 3 = 2 m/s2, not 8 / 3.
- Writing m/s for the unit. m/s is the unit of velocity. Acceleration is in m/s2.
- Mixing up fast and accelerating. A fast object at a steady velocity has zero acceleration.
- Losing the minus sign. If an object slows down, the acceleration must come out negative.
Exam-style question
A lift in a tall building starts from rest. It reaches a velocity of 6 m/s in a time of 4 s.
(a) State the formula linking acceleration, change in velocity and time taken. [1 mark]
(b) Calculate the acceleration of the lift. Give the unit. [3 marks]
(c) Near the top of the building the lift slows down with an acceleration of −2 m/s2 until it comes to rest. Its velocity before it starts to slow down is 6 m/s. Calculate the time taken for the lift to come to rest. [3 marks]
Model answer
(a) acceleration = change in velocity / time taken, or a = (v − u) / t (1)
(b) a = (6 − 0) / 4 (1)
a = 1.5 (1)
unit: m/s2 (1)
(c) t = (v − u) / a (1)
t = (0 − 6) / −2 (1)
t = 3 s (1)
Exam tip
Before you calculate, write a list: u = , v = , a = , t = . Fill in what the question gives you, turning "starts from rest" into u = 0 and "comes to rest" into v = 0. The gap in your list is the quantity you need to find, and the list makes it hard to put a number in the wrong place.