A **displacement function** tells us how far a particle has moved from a starting point (an origin) at an given time.

Note that **displacement is not the same as distance traveled**; while a particle might travel back and forth or in circles, the displacement only represents the difference between the starting and ending position. It is a vector quantity, which means it has both a value and a direction (e.g. 20 m north or minus 50 feet). It is only equal to the distance traveled if the motion is straight line in a constant direction.

## Finding the Displacement Function in (Relatively) Simple Situations

**Example question:** Suppose a ball is traveling with an initial velocity of 11 m/s and a acceleration of 2 m/s down a slope. What is the displacement at t = 2?

The formula to find the displacement (Δ s) is:

Step 1: **Identify the parts of the equation** (from the question):

- v
_{0}= 11 m/s - t = 2 s
- a = 2 m/s

Step 2: **Insert the values into the formula and solve.**

By the above equation, the displacement at t = 2 will be:

( 11 m/s · 2 s) + (½ · 2 m/s · 2 s^{2}),

Giving the answer as **22 m + 4 m = 26 m. **

## How is the Formula Derived?

Let’s say a two-dimensional particle is undergoing motion in some constant direction (imagine a dot on a number line).

The speed of the particle can be changing, but to make the math easy, let’s say that the acceleration is constant.

Acceleration is defined as the *change in velocity per unit time*, so if you have a starting velocity and a constant acceleration, you can write the acceleration at any given time in terms of the velocity:

or

Using a little algebra, you can write the velocity in terms of the acceleration:

Velocity is the change in position per unit time, so the velocity of a particle is the first derivative of the position.

and

With a little rearrangement and algebraic substitution, you get:

Setting the initial distance to 0, you get an equation which gives you the displacement, or change in position.

## References

Distance, Displacement and Position. Retrieved from https://washingtonlee.apsva.us/wp-content/uploads/sites/38/2017/02/Distance-Displacement-and-Position-Notes.pdf on Feb 26, 2019

Elert, Glenn. Equations of Motion. The Physics Hypertext. Retrieved from https://physics.info/motion-equations/ on Feb 27, 2019.