Lab 3: March 8,2017: Non-Constant Acceleration Problem

Lab 3: Non-Constant Acceleration Problem

Amy, Chris, and John

March 8,2017

Today's lab was trying to find the distance an elephant wearing roller skates with rocket strapped to its back with a non-constant acceleration would go before coming to rest with derivations and excel spreadsheets. By shortening the intervals of time where average acceleration of an interval equals instantaneous acceleration then we can derive the change in position. 

Theory: For this lab our group was trying to analyze when an elephant, wearing roller skates with a rocket strapped to its back, would come to a stop before rolling off a cliff. The elephant has a rocket strapped to its back serving a deceleration that counts as a loss of mass as it continues burning fuel.

Apparatus and Procedure: For the equipment in this lab it was a laptop with excel and our lab handout. First we solved this problem derivations analytically. We were given a function of acceleration in respect to time which we integrated to find a function of time and integrated again to find a position function. This process was provided to us in our lab handout as well.

Derivation for our time

Once the position function is found then we can figure out how far the elephant will go before it comes to rest, in order to do that we need to find the time the velocity of the elephant is zero.



Professor Wolf told us that this process of integration is arduous and said that this analytical approach could be very time consuming and we have to learn to recognize that some functions cannot be integrated at all. For our lab our acceleration function was able to be integrated but a numerical approach can be used as well using excel.

With our lab handout we were able to plug numerical values given to us into a spreadsheet in excel and with better precision and much faster solving, excel came up with a solution. Because of excel's capability of calculating and manipulating cells we were able to get the value of distance much faster than integrating and plugging in for our values.

Measured Data: While using the analytical approach our value for time was 19.69075 seconds and our distance was 248.7 meters to prevent the elephant from rolling off the hill.

First spreadsheet using intervals of 1 second

Second spreadsheet with intervals of 0.1  seconds

Third spreadsheet with intervals of 0.05 seconds

As we get smaller with the time intervals we notice that the change on velocity from a positive to negative number is much more accurate and noticeable. In our first spreadsheet the change was much more difficult to find while with the smaller intervals like in spreadsheet 3 we see the velocity get much smaller before becoming a negative number. In our analytical calculations we calculated that the elephant's velocity turns to zero at 19.69075 seconds. In spreadsheet 3, we see that the change from positive to negative v happens at 19.65 seconds to 19.7 seconds which supports our numerical value. As the time intervals get smaller we see that the answers are becoming more precise and with enough intervals one might approach the same solution of 19.69075 seconds for the velocity to equal zero. 

Conclusion: Our group was able to see that the two approaches are accurate and can get us virtually the same answer. Although the numerical approach was much faster, we saw that without an analytical approach we would not have anything to compare to. Both approaches might be necessary to do this experiment in order to compare numbers. This lab was a good way of becoming acquainted with excel and getting more used to using other tools than work in faster and accurate ways. Due to excel's much faster way of calculating we would resort to using excel more often during our lab calculations. 

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

-For our analytical calculations we got much more precise numbers. The time we got was 19.69075 seconds, which is attainable with our numerical approach although it might take some time to divide the time intervals in order to get that many decimal points. The distance we calculated was 248.7 meters and excel figured a much more precise number of 248.6976 meters with much less uncertainty, one could say that both methods are accurate. Each method having a very small uncertainty because we did not do many calculations where we propagated our uncertainty. 

2. How do you know when the time interval you chose for doing the integration is "small enough"? 

-Without the analytical result we could still break down time intervals in our excel spreadsheet until our velocity equaled zero. When velocity equaled zero the elephant had come to a stop providing us with the distance and as we continue to make intervals smaller then we could get as accurate result as the analytical calculations. 

3. Determine how far the elephant would go if its initial mass were 5500 kg, the rocket mass is still 1500 kg, but now the fuel rate is 40 kg/s and the thrust force is 13000 N. 

With the new variables given, we can still calculate the distance. By replacing the mass, the rocket mass, the fuel burn rate, and the thrust force we get a position of 164.036268 meters at the time interval 12.95 seconds to 13.0 seconds.