Lab 3: Non-Constant Acceleration Problem 3/4/15

Physics Lab #3: Started on 3/4/15
Non-Constant acceleration problem/Activity problem
Annemarie Branks
Professor Wolf

Objective: Find how far the elephant travels before coming to rest, analytically and numerically. Be able to explain how you can find the answer numerically.

Procedure:
     1.  After dividing the net force by the changing mass equation + the mass of the elephant to find acceleration, we would want to used calculus to to find the position equation. This method would result in long, extensive calculus that is prone to human error. We will determine how far the elephant travels before coming to rest by setting up an Excel spreadsheet to assist us in determining the position.
     2.  In the first column we list time in whatever increments we choose. The smaller the increments, the more accurate our answer will be, but the increments only need to be so small before the changes begin to be insignificant. The next column we enter in our acceleration equation which ended up being -400/(325-t) m/s^2.
     3.  Make a column showing the average acceleration of each time interval. We need to determine the average acceleration because this will allow us to determine the change velocity. Average acceleration multiplied by the chosen Δt will give the change in velocity between two specific times.This can also be thought of as looking at the acceleration curve, drawing the curve for each time increment as a straight line, and finding the area under the curve. Δv = Δt(a1+a2)/2 
     4.  Now that we have found the change in velocity, we can find the specific velocity for each point in time. Since the elephant is starting out with 25 m/s the following time can be determined by adding the change in velocity between 0 seconds and the next time with which you chose to put your interval at. Use the Excel functions to continuously add the Δv to the previously determined velocity until the velocity reaches zero.
     5.  Using the same method we did to determine velocity, we are going to find position. Make a column listing the average velocities for each time interval and in the next column list Δx by taking the average velocities and multiplying them by Δt. This is also like looking at the velocity curve and calculating the area under the curve to find position. Δx = Δt(v1+v2)/2
     6. Similar to how we found velocity, we will find position by adding the change in position to the the position at the beginning of the interval to find the position at the end of the interval. Use Excel functions to repeat this process for each time interval until you reach the point where velocity became zero. At the time is where you will find the elephant's position when it stopped moving.

Results:
We decided that a Δt of 0.01 seconds was small enough. The velocity reached zero at t=19.69 seconds where its position was 248.698 meters. This value matched with the answer we calculated using integration which was 248.7 m. Therefore, we can conclude that this method is a great alternative to integrating difficult problems. This method also showed us that the smaller Δt is, the more accurate our answers are. Calculus would assume that Δt is infinitely small which is the most accurate but the way we did it on Excel is as good as we would want it to be.

This is our spread sheet where the cell for the velocity is closest to zero. All the way to the right in column H shows the position of the elephant at this time.