September 7, 2016: Non-Constant Acceleration Activity

Anthony Betancourt
Lab Partner: Josh Fofrich
Professor Wolf

Lab #3: Non-Constant Acceleration

Purpose: 
The purpose of this activity is to utilize Excel in achieving a numerical approach to solving  non-constant acceleration problems.  Analytical approaches to problem solving can be time consuming and messy, therefore, plugging values into Excel can calculate large amounts of data quicker and more efficiently.  We are giving a problem of an elephant with a rocket strapped to its back, headed down a ramp at a given velocity.  The moment the elephant reaches the level surface at the bottom of the ramp, the rocket ignites and begins to thrust in the opposite direction that the elephant is traveling.  We need to find out how far the elephant travels before coming to rest.  

Procedure:

1. Open up a new Excel spreadsheet
2. Fill out the correct values for Vo and Xo
3. Set delta t to be 1 second. Put in the other values in cells B1 through B4
4. Input a formula into cell A9 that can calculate time, and fill down.  
5. Enter an acceleration formula into B8 that will calculate acceleration at any given time.  
6. Cell C9 enter formula to calculate the acceleration for the first interval of time.  
7. Cell D9 enter formula to calculate change in velocity
8. Cell E9 enter formula to calculate the speed at the end of the time interval.
9. Cell F9 enter formula to calculate the average speed at the time interval
10. Cell G9 enter formula to calculate the change in position at that time interval
11. Cell H9 enter formula to calculate the position of the elephant. if done correctly you should be able to fill down contents of row 9 and see at what distance the elephant comes to rest at.  Answers should agree closely with what you solve analytically.
12. Change time interval from 1s to 0.1s and take note of any difference made.
13. change time interval from 0.1s to 0.05s and take note of any differences.  

Theory: 
The use of Excel provides us a way to numerically represent each second of the elephants movement until it reaches a resting point.  In order to verify our analytical method, it would be easy to double check by plugging in values generated by the spreadsheet.  In cases where it is time consuming or not possible to be performed analytically, Excel is a powerful tool that can be used to make light work of repetitive data plugging.  

Measured Data:

(Excel table filled out with given values from procedure)

(1 second intervals; H28 shows distance traveled before coming to rest)

(0.1 second intervals; H205 shows distance traveled before coming to rest)

(0.05 second intervals; H402 shows distance traveled before coming to rest)

Analysis:  

The Excel tables above depict the distant the elephant traveled until coming to rest in increments of 1 second, 0.1 second, and 0.05 second intervals. In order to properly depict the data, certain equations must be enter into the appropriate cells.  Cell C9 is "=(B8+B9)/2", D9 is "=((C9*$B$4))", E9 is "=(E8+D9)", F9 is "=(E9+E10)/2", G9 is "=F9*$B$4", and H9 is "=H8+G9".  As the time interval decreases, the more precise the value for the distance becomes. 

Conclusions:    

1. Analytical results are dependent on your calculus skill and the amount of time it takes to solve the problem, Whereas, numerically solving the problem using Excel can decrease the amount of time and the likelihood of systematic errors.  
2. Choosing the appropriate time interval to show the correct integration result depends on how exact of a value needed for the distance traveled, the small the interval the more exact of a value for delta x.  Excel is beneficial in this instance because the data table can be adjusted to calculate smaller and smaller values for time and result in a more accurate value for the distance.  
3.   The elephant would travel approximately 164.036268 meters before coming to rest.