# Inquiry: **Calculus of Motion**

## Purpose: To examine and apply calculus to the motion of objects.

## Procedure:

The velocity at a given instant is called the instantaneous velocity. It is the magnitude of this instantaneous velocity that we read on a speedometer. Also it is the instantaneous velocity that we can calculate by using the slope method of a graph of distance vs. time. We use the word instantaneous to distinguish the velocity at one moment from the average velocity over an interval of time. In this Inquiry we will be graphing uniform acceleration on a D-T, V-T and an A-T graph to view how they are related.

The photo gates are connected in series. When switch one is activated, the timer will start to run and when switch 2 is activated the timer will stop. Switch one has been mounted at the beginning of the track and you will leave it there throughout this Inquiry. Place the second switch 0.2 m away from switch one for the first trial and then increase the distance by 0.5 m intervals until you have the time interval over the distance of 4.2 m and record in your notebook. Make all runs by placing the ball against the cork on the high end of the track.

## Inquiry Questions:

- Use a spreadsheet to make a D-T graph with distance on the y-axis and time on the x-axis and put on a polynomial trend line to get the equation. Take the derivative and use this equation to make a V-T graph. Take the derivative of this line to get the constant acceleration and then make an A-T graph. Now calculate the area under the A-T graph at some time and compare to the velocity on the V-T graph at this same time. (You will have to add on any initial velocity) Now calculate the area under the V-T graph at the same time and compare to the displacement on the D-T graph at that time. Print all graphs and insert into your notebook.

- If the position of a uniformly accelerating object is given by 4t
^{2} + 7t + 1 what would be the velocity of the object when time is at 5.0 s?