March 04, 2015 Non-Constant Acceleration Problem

Purpose:
To solve an integration problem using microsoft excel by following a certain logic and inputting corresponding formulas.

Problem:

We may solve this problem by integration. Which is shown below with detailed steps. The strategy is firstly to find the acceleration of the elephant/rocket system. Secondly, use a(t) to find v(t) by integration. In the same way, use v(t) to find x(t) by integration. Finally, find the time it takes for the elephant to stop then plug the time into x(t) to get the distance. The distance is 248.7 m.

OR

This is not all, this is just a sample of the spreadsheet 

We can use excel spread sheet to come up with the same distance.

1.) Take 8 columns and named them as the following: t (time), a(acceleration), a_avg (average acceleration), ∆v (change in velocity), v (velocity), v_avg (average velocity), ∆x (change in distance) and x (distance).

2.) Set the time increments by 0.1 seconds for many rows. We began with time because all the other components depends on time.

3.) Input the formula for acceleration column. The formula is the function of acceleration with respects to time, which is a(t)= -400/(325-t).

4.) After getting the acceleration for each time increment, find the average of the acceleration in the next column. The reason we find average acceleration is that average acceleration times time results in the change in velocity.

5.)  For the change in velocity column, multiply the average acceleration by time. The change in velocity helps us to find the final velocity of that time increment. The final velocity is the initial velocity plus the change in velocity.

6.) We form the velocity column by adding the every velocity, starting with the initial velocity of 25m/s, to the change in velocity.

7.) Then we take the average of the velocity column. The average velocity is needed to find the change in distance.

8.) The final distance is the sum of the change in distance

To find the distance when the elephant stops, find the time when it stops from the integration calculation, which is 19.7 seconds, then look at the distance at the end of the row, which is 248.929 m.

Conclusion:

1. The distance we get from doing integration by hand is 248.7 m while the distance from the excel spreadsheet is 248.9 m. The reason for the difference in the final distance is because the integration by hand uses infinite amount of rectangles while the excel spread sheets with time increment 0.1 uses only 200 increments (rectangles). 

2. When the figures in the average acceleration column does not seem to change much, then you know you have the right increment of time. This is because when we use a larger time increment such as 1 second, the average acceleration graph is a curved line, we cannot integrate. However, when we use smaller time increment such as 0.1 second, then we break the same curve into smaller piece, which now we can look at them as little straight line for integration.