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):
- v0 = 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 s2),
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).
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:
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.
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.
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.
Stephanie Glen. "Displacement Function: Definition" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/types-of-functions/displacement-function/
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