Acceleration

Acceleration is the rate that an object’s speed or velocity is changing. In physics it is measured in meters per second squared, or ms^-2.

The basic formula for finding the average acceleration is this:

By the way, the triangle stands for "change of". In this case we need the change of velocity.

Remember to convert the measurements into their proper SI UnitsA list of these can be found in the Before We Begin section. first.

When calculating the change in speed, you may get a negative answer. This is fine; it just means there is a deceleration. See Example 2 for more info.

Another thing to remember is that because acceleration is a change in velocity, it can come with a direction. So if an object is traveling around a corner at a constant speed, it would be "accelerating" because even though the speed is the same, the velocity is changing. Hard to digest, but we'll try and explain it again a bit later.

That's right, it’s already time for examples!

A car goes from 5ms^-1 to 17ms^-1 in 4 seconds. What is its acceleration?

So how do we find the change in velocity?

To find the change in something, it is the final amount minus the initial amount.

In this case, the final velocity is 17, and the initial velocity is 5.

So the change in velocity is 17-5, which is 12.

Back to our original formula, we can now find out the acceleration.

a = 12÷4
a = 3ms^-2 (1 s.f.Don't know what this is? Significant figures are explained in the Before We Begin section.)

And that’s all there is to it.

This formula can also be used for decelerationThis is another word for slowing down. .

A marble traveling at 2ms^-1 rolls along a carpet and slows to a stop after 1 second. What is its acceleration?

We know that t = 1, but what is the change in velocity?

Remember that the change in velocity is final-initial, so it's 0This comes from the fact that the marble stopped, i.e. it is going 0ms^-1.-2, which is... wait a minute... negative 2?

A negative answer? Something must be wrong…

Don’t Panic.

Check the Danger section. The negative sign means it’s a deceleration.

So…
a = -2÷1
a = -2ms^-2Remember, this is the acceleration, like the question asked for. If it asked for the deceleration, we only need to reverse the sign, so it would be 2ms^-2. (1 s.f.)

Before we move on, we'll look at vectorsIf you don't know what a vector is, check out the Displacement section. again to explain the second point made in the Danger section.

Looking at the formula near the top of the page, we need to find the change in velocity in order to find the acceleration. However to do that we need to subtract the two speeds, which we will draw as vectors. Only one problem though, how do you subtract a vector? Hmm...

To subtract a vector, we add the negative version of a vector instead; giving us v₁ + ^–v₂.

Let's test this with an example.

A car travels at 5·0ms^-1 to the right, then changes direction and goes 5·0ms^-1down in 7·0 seconds. What is its acceleration? Now, even though the speeds are the same, the object still has an acceleration because the direction is changing.

Lets look at this as a diagram.

The first picture is the trip that the object in question took, and you could also express it as v₁ + v₂. But to find the change in velocity we need to turn the first 5ms^-1 measurement around to make it negative, as explained above.

After we do that, the diagram looks like the second picture. Using amazing tools of trigonometry will help us find the answer. First we use the Pythagorean theoremRemember this from Year 11? It's a²+b²=c². to find the length of the change in velocity like so:

5²+5² = x²
25+25 = x²
x² = 50
x = 7·1ms^-1 (1 s.f.)

Taking that speed we divide by the amount of time taken to find the acceleration.

a = v ÷ t
a = 7·1 ÷ 7
a = 1·0ms^-2 (2 s.f.)

Now we find the angle using SOHCAHTOA. Because we know the opposite and adjacent side lengths, we will use the "tan" function as follows:

angle = tan^-1(5÷5)
angle = tan^-1(1)
angle = 45°

Now we take the two parts of the answer and put them together to make...

1·0ms^-2 at 45° (2 s.f.)

This method of finding the answer can also be used if the two speeds are different sizes.

Similar problems can also be found near the bottom of the Displacement and Velocity pages.

Got Acceleration under your belt? Have a look at Speed with Acceleration, Projectile Motion, or Circular Motion.