PHYS 4A/ Fall 2014; Rocket Powered Elephant

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

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

In order to solve the problem analytically, the following acceleration equation was formed.

The acceleration equation was then integrated from a time=0 to time=t in order to find the change in velocity. 

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

Position can be solved by determining initial velocity and using that value in order to the position function. This will give a distance value of how far the elephant will have gone,

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.

Newton's Second Law states that Force=mass*acceleration (F=ma). Thus 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.

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 set 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. As shown in the highlighted region, the velocity is zero between 19.6-19.7 seconds with position being around 248.7 meters.

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. Depending on the level of precision needed for an experiment, the intervals can range anywhere from years to a fraction of a second. 

When analyzing data, it is more efficient to do things numerically due to the speed of calculations and the huge amount of data that can be presented. When solving analytically, it would be very difficult to predict how the object may have been moving in between intervals and calculations would be very tedious. By using a spreadsheet software such has excel that can do many calculations instantly, precision can be based on the needs of the experimenter. 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.