Non-Constant acceleration problem/ Activity

Non-Constant acceleration problem/ Activity:

Introduction:
In this activity, we are trying to solve a problem that involves an object moving with a non-constant acceleration:

Using Newton's second law, we can set up an equation for the acceleration of the elephant:

a(t)= Fnet / m(t)= -8000 N/(6500 kg-20 kg/s*t)= -400/(325-t) (m/s^2)

Although we can integrate a(t) to get functions of v(t) and x(t), the result of integration would be very complicated and very hard to solve. 

In this activity, we learned to use the software spreadsheet to solve complex problem like the one above.

We first set up the first few rows of Excel to simulate the movement of the elephant in tiny time intervals, and shows the acceleration, average acceleration, difference in velocity, velocity, and distance traveled in different columns. Initially we set the time interval to be 0.1 second, and fill the whole rows down to obtain data for about 220 rows. After doing so, the result looks like the following picture:

The velocity of the elephant when it comes to stop will be 0, and to find the distance traveled by the elephant before it stopped, we simply have look at the row at which v=0:

According to the chart we made, the elephant stopped between time interval 19.6 second and 19.7 second after the motion begins, and thus, we can conclude that the elephant traveled 248.695 meters before came to stop.

Questions:

1. Compare the results you get from doing the problem analytically and do it numerically.

   To do it numerically, we derived the function a(t) to get function of velocity v(t) and function of distance traveled x(t):

Then, we look for the value of t when v(t) = 0, and it turned out that our answer falls at t = 19.69 second, which is the same as the time we acquired from Excel.

2. How do you know when the time interval you chose for doing the integration is "small enough"? How would you tell if you didn't have the analytical result to which you could compare to you numerical result?

If the time interval is not small enough, velocity will never be very close to zero, but change from a positive number to a negative number suddenly. If the time interval is not small enough, the answer we found will not be precise enough to tell how much distance the elephant really traveled before coming to stop.