04-March-Non constant acceleration problem

Rocket Powered Elephant

Purpose

To use excel to determine numerically how far an elephant goes before coming to rest.

This experiment is essentially used to determine how far an elephant would move before coming to rest. The distance traveled was found by using both analytic and numerical integration.

Analytic Solution

In order to solve the problem analytically, we used the following acceleration equation (Newton's 2nd law)

We then integrated this acceleration equation  from  time=0 to time=t in order to find the change in velocity. 

We then integrated the velocity equation using the same time parameters as the previous integration in order to obtain a position function relative to time.

To solve for position we determined the initial velocity and the time at which the velocity of the elephant is zero. Plugging this values in the position function relative to time gave us the distance traveled by the elephant.

Numerical Solution

In order to solve the problem numerically,a table like the one shown below was created in order to organize the data and make use of excel's ability to take a formula and apply it to other cells within the spreadsheet.

How to Set up the columns: 

Newton's Second Law states that Force=mass*acceleration (F=ma). Thus as seen above in order to find acceleration, Force can be divided by a mass. In the scenario that was given, a rocket produced 8000N of thrust onto a total mass given as a function of time m(t)=6500kg-20kg/s*t. This equates to acceleration being -400/(325-t) (m/s^2). By using this value for acceleration as a function of time and inserting it into the excel sheet, acceleration at any given time was calculated as seen in the second column. 

Using that information, we were able to calculate the average acceleration between every time interval which was put into column 3. Change in velocity was calculated using the average acceleration within a certain interval divided by the change in time. Individual velocities at any given time were then found by taking the initial velocity of 25m/s and adding the change in velocity. The change in position between intervals was determined using a kinematics equation, taking the average of velocities within an interval and multiplying that by the time passed. Finally, position from the origin was found by adding the changes in position between intervals.

What the columns looked like:

After doing these calculations, a data table of over 800 points was produced in order to see changes in data if the time intervals changed. The table shown above had a time interval of 0.1 seconds and shows the first 2 seconds of acceleration as an example.

Below is the spreadsheet when velocity reaches 0 using 0.1 as a time interval. As shown in the highlighted region, the velocity is zero between 19.6-19.7 seconds with position being around 248.7 meters.

However, If the intervals are adjusted, you are able to determine the exact moment where velocity reaches zero more precisely. Below the intervals were adjusted to 0.05 seconds. In the highlighted region, velocity is zero between 19.65 seconds and 19.70 seconds. By making the intervals smaller, the values of time become more precise. 

Conclusion

To analyze date, its better and more efficient to do it numerically instead of analytic because of the huge amount of data than can be processed in a matter of seconds. When doing things analytic is hard to know how an object may have been traveling in between intervals of time and is also tedious because integrating is usually difficult.

When making the intervals very small, there may be a point where values start to become constant; it may be at this point the experimenter may feel that they do not need to go any smaller in precision. If solving analytically, the precision would have to be based solely on significant figures and the power of the calculator that is being used.