What is acceleration?

This page reviews the definition of acceleration in one dimension. For a longer discussion, see your friendly local textbook.

Here is a list of equations useful for constant acceleration in one dimension, and a discussion of how to use them to solve problems.

Position and related quantitites

Position

Think of an object as being located at one point along a straight line. Suppose we call one point on the line the "origin", and pick a direction on the line to be the positive direction. Now we number all the points on the line (like the number line) according to their distance (in some sensible unit like meters) and direction from the origin, with points in the positive direction from the origin getting positive numbers, and points in the negative direction getting negative numbers. Then we can give the position along the line as a number. Let's call that number x. (We could use any letter.) So, for example, if the object is at the origin, x is 0. If the object is two units in the positive direction from the origin, x is 2.

Displacement

If an object moves from position x_1 to position x_2, its change in position is x_2 - x_1. This change in position is called "displacement".

Notice that displacement can be either positive or negative. If the object moved towards the positive direction, the displacement is positive. If the object moved towards the negative direction, the displacement is negative.

"The Change In"

A few comments about how physics uses the phrase "the change in ". If some quantity such as x has a starting value and later has a final value,

(The change in x) = (The final value of x) - (the starting value of x).

We will do this with other quantities than x. "The change in" is used so much in physics and some parts of math that it has a special abbreviation Δ.

So we can write:

(The change in x) = (The final value of x) - (the starting value of x) = Δx.

And, if x is position,

Displacement = Δx.

Distance

The distance between two points x_1 and x_2 on our line is the absolute value (magnitude) of x_2 - x_1. Using the absolute value symbol we can write:

distance = | x_1 - x_2 |.

Similarly, if an object moves from x_1 to x_2, it moves through a distance of | x_1 - x_2 | = |displacement|.

Unlike displacement, distance is always positive.

In physics, we take the attitude that displacement is a more fundamental thing than distance.

Velocity and speed

In physics, a thing can be at one place at one time and at another place at another time.

Average velocity

Suppose an object is at position x_1 at time t_1, and at position x_2 at time t_2. Then, according to what we said earlier, its change in position is

Δx = x_2 - x_1.

Perhaps not as obviously, the time interval t_2 - t_1 over which this change in position happens can be written as

Δt = t_2 - t_1.

Then we define the average velocity over that time interval to be:

(average velocity) = Δx / Δt.

Velocity

Suppose we find the average velocity in smaller and smaller time intervals, all of which contain some time t. If there is some value that the average velocity gets closer and closer to as the interval around t is made smaller and smaller, that value is called the "instantaneous velocity" of the object at time t. When we just say "velocity" we will mean the instantaneous velocity. Typically we will use the letter "v" to stand for velocity.

In English, velocity is how fast position is changing.

If we graph position x versus time t (that is, with the vertical axis being x, and the horizontal axis being t), the slope of the graph of x(t) is the velocity.

(For those who know calculus, this can be written as

v = dx/dt.)

If we graph the velocity of an object versus time, we can talk about the area between the curve and the horizontal axis. If we find this area between time t_1 and time t_2, it turns out to be the object's displacement between time t_1 and time t_2. When doing this, the area above the horizontal axis counts as positive displacement, and the area below the curve counts as negative displacement.

Speed

Speed is the absolute value (magnitude) of velocity. Thus, if velocity in one dimension is v, the speed is |v|. In physics we consider velocity to be a more fundamental thing than speed.

Acceleration

The velocity of an object at one time can be different from its velocity at another time.

Average acceleration

Suppose an object has velocity v_1 at time t_1 and velocity v_2 at time t_2. Then its change in velocity is:

Δv = v_2 - v_1.

and, as above, the time interval is Δt = t_2 - t_1.

Then we define its average acceleration over that time interval as

(average acceleration) = Δv / Δt.

Acceleration

Suppose we find the average acceleration in smaller and smaller time intervals, all of which contain some time t. If there is some value that the average acceleration gets closer and closer to as the interval around t is made smaller and smaller, that value is called the "instantaneous acceleration" of the object at time t. When we just say "acceleration" we will mean the instantaneous acceleration. Typically we will use the letter "a" to stand for acceleration.

In English, acceleration is how fast velocity is changing.

If we graph velocity v versus time t (that is, with the vertical axis being v, and the horizontal axis being t), the slope of the graph of v(t) is the acceleration.

(For those who know calculus, this can be written as

a = dv/dt.)

Notice that this definition is analogous to the definition of velocity.

If we graph the acceleration of an object versus time, we can talk about the area between the curve and the horizontal axis. If we find this area between time t_1 and time t_2, it turns out to be the object's velocity between time t_1 and time t_2. When doing this, the area above the horizontal axis counts as positive velocity, while the area below the curve counts as negative velocity.

Answers to Sample Conceptual Questions

Question 1: Position is to velocity as velocity is to what?

Answer: Velocity is how fast position is changing, while acceleration is how fast velocity is changing.

So the answer is acceleration.

Question 2: Displacement is to distance travelled as velocity is to what?

Answer: Distance travelled is the magnitude of displacement. Speed is the magnitude of velocity. So the answer is speed.

Question 3: I see that speed is the magnitude of velocity. Is there any word that means the magnitude of acceleration?

Answer: No. But mathematically, you can write |a| for the magnitude (absolute value) of an acceleration a.

Question 4: If velocity is how fast position is changing, and acceleration is how fast velocity is changing, is there any word for how fast acceleration is changing?

Answer: Yes, engineers use the word "jerk" for how fast acceleration is changing.

Question 5: Okay, so far we have Position, Velocity, Acceleration, Jerk ... Can this process of defining new quantities, with each being how fast the previous quantity is changing, go on forever?

Answer: Yes. However, only the first few such quantities have names. After jerk, the next three quantities are sometimes called "snap", "crackle" and "pop". These names started as a joke; they were taken from the commercial for Rice Krispies (TM).