# How To Calculate Acceleration

By the end of the 16th century, Galileo Galilei allegedly demonstrated that all objects on Earth will fall down at the same speed when thrown from the same height regardless of their mass or size. He did this by simultaneously releasing two cannonballs of different sizes from the Leaning Tower of Pisa, and showing they reached the ground at the same time.

For people at the time this was a crazy idea, since intuition would make most of them expect the heavier mass to fall down faster than the lighter one. What Galileo was proving was that, even a feather and a cannonball, when thrown simultaneously, would fall at the same speed, provided no friction with the air is present.

On the other hand, when comparing racing cars, it is very common to use their 0 to 60 times. This is a test made on specialized racing tracks, where the time it takes a car to go from 0 mph (miles per hour) to 60 mph is measured using very precise stopwatches.

So, what do these two scenarios have in common? The answer is acceleration. What explains the Galilean experiment discussed above is that all objects on the Earth are subjected to the same gravitational acceleration, which arises from the gravitational attraction between our planet and their masses. On the other hand, in 0 to 60 measurements, it is precisely cars’ average accelerations what is compared indirectly at the end.

Acceleration is therefore a very important concept when dealing with moving bodies. Let’s discover what it is and how we can measure it.

## How to calculate acceleration

To calculate a body’s average acceleration follow these steps:

1. Subtract the initial speed from the final speed of the body, v.
2. Determine the time it took the body to go from the initial to the final speed, t.
3. Use the following equation to calculate the body’s average acceleration:

The average acceleration vector is parallel to the velocity vector. Its sense depends on the sign of the resulting value: if positive, the vector points in the same direction as the velocity vector; if negative, it points in the opposite direction to the velocity vector.

## What is acceleration?

Velocity tells us the rate at which movement occurs. This means, it relates the displacement of an object to the time needed for it to experience said displacement. Since no sudden changes in position are possible, at least in classical mechanics, all displacements require a certain amount of time to take place. How both quantities relate is precisely what velocity describes.

Average velocity is calculated using a moving object’s displacement vector. If you are not familiar with this concept, go ahead and read our article about it. Because of the definition of displacement, the object’s final change in position is what is considered in the calculation. To do this, we compare the object’s position at the beginning and the end of the time frame for which we wish to calculate its average velocity:

Here, x is the body’s displacement and t the time difference between the initial and final points of its trajectory, which equals the time it took for it to experience said displacement. If you want to explore the concept of velocity in depth, we invite you to read our article about it.

Velocity is a vector. This means, it has both a magnitude and a direction. The former is called speed, and it tells us how fast the object is moving. The latter, according to equation 2, equals that of the object’s displacement vector.

The definition of average velocity ignores all possible changes in both its speed and its direction that occur within the selected time frame for the calculation. Imagine you take a car and drive to a friend’s house. When parked, the car has a speed of 0 km/h. Once you start driving, you gradually increase velocity until a certain value, let’s say, around 50 km/h. Then, if you find a stop light, you will decrease your speed back to 0 km/h. Even when turning, you should first decrease your speed and then increase it back to your desired cruise value.

Nevertheless, if you calculate your average velocity, going from your house to your friend’s house, the result would only take into account the distance along a straight line between the starting and ending points, and the total time it took you to cover said distance. All velocity changes, both in speed and direction, are ignored, as the following image shows:

We have established that velocity changes may occur along a moving body’s trajectory. Now, given that sudden changes in position are simply not possible, because velocity increases gradually and not in “leaps” from one value to another, those velocity changes can’t take place suddenly either. Let ‘s discover why.

Imagine you want to take a car from 0 mph to 60 mph. To do so, you need to press the gas pedal. Let’s imagine you do it gently, so the car starts to increase its velocity gradually until you reach 60 mph one minute later. Now, imagine you repeat this experiment but decide to press the gas pedal at once to its full extent. In this case, the car will certainly react with more power as before, and will probably reach 60 mph in a shorter amount of time, let’s say 30 seconds.

Both scenarios can be seen in the following graph. The blue curve shows how the car’s speed increases by gently pressing the gas pedal, while the red one shows how it increases when pressing it fully. In both cases, the total velocity increase, namely from 0 mph to 60 mph, takes some amount of time to be completed.

Acceleration measures how fast velocity changes occur, and it therefore relates any change in this variable (in magnitude, direction or both) to the time needed to experience or execute said change, or more concisely, the change in velocity per unit time.

It is calculated by dividing the difference between the initial and final velocities, v, by the time difference between the moments in which said values take place, t, as equation 1 shows. Keep in mind that the symbol (a capitalized greek delta) represents the change in a variable between an initial and a final state. It is defined as the final value minus the initial value. Since only these two values are considered and any other values in the middle are ignored, this calculation yields the average acceleration.

In our previous example, in the first scenario, the car’s velocity went from 0 mph to 60 mph in 60 seconds. This means the car’s total acceleration within that period of time was:

Since the result is positive, and according to our definition of the velocity difference (v), it means the final speed had to be greater than the initial one. This indicates that the car’s velocity increased during the selected period of time. If the result were negative, that would mean the car’s speed would have decreased in that time slot, meaning it was braking. This can be represented as the acceleration vector acting in a direction opposite to that of the velocity vector. In the second scenario, the car’s acceleration was:

The difference between both scenarios is that the car’s acceleration was greater when the gas pedal was fully pressed. This is precisely why it took the driver a shorter time to reach the goal speed (60 mph) in the second case.

This way of determining the power of a car is called measuring its “0 to 60 time”, which is used to compare its performance to that of other cars, especially in professional car racing.

## Instantaneous acceleration

Notice that in our example, the car’s speed increases uniformly from 0 to 60 mph. This could not have been the case, though. For example, you could have decided first to press the gas pedal slightly until reaching a certain speed, and then pressing it fully to reach the final value. This scenario could look something like this:

Here, the red circles represent points at which the acceleration changes. Keep in mind that velocity is always changing within the first 60 seconds and only remains constant afterwards. The acceleration, on the other hand, only changes twice: at t = 30 s and at t = 60 s.

In this case, the car’s acceleration was not the same along the entire trajectory. Nevertheless, if we calculate it as shown before we would obtain the same value as in our previous example:

This occurs because the way we calculated the car’s acceleration only takes into account the final and initial speed values, which is why we call this the average acceleration. Nevertheless, as we just showed, a moving body’s acceleration can change within a single time frame, even to a point where it is constantly changing. Look at the following graph, the more acceleration changes there are, the closer the velocity vs. time plot gets to a soft curve.

So, if we want to know the value of acceleration at a single point, instead of an average value for the entire trajectory, we would need to subdivide it into many small sections, for which the time frame becomes smaller and smaller. This is done, mathematically, using a limit. In this case, that of the definition of the average acceleration, as t tends to zero:

Here, v(t) is just a reminder of the velocity being a function of time, which means it changes with time. Equation 6 is the definition of the derivative of the velocity with respect to time, which is also called the instantaneous acceleration

To calculate the instantaneous acceleration, a function describing the body’s velocity in terms of time as an independent variable is needed.

Use this simulator by PhET to observe how the velocity and the position of a walking man changes as you set initial values for them and for his acceleration. Keep in mind you can use positive or negative values to specify the sense of the velocity and acceleration vectors. Go to the “Charts” tab to see the resulting position vs. time and velocity vs. time plots.

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