Calculating Non-Constant Acceleration problems. Jose Rodriguez; Lab partners: Kevin Nguyen; Kevin Tran: Lab Date: 3/8/17

This experiment was intended to teach the students how to solve a non-constant acceleration physics problem with the help of Microsoft Excel in a numerical style approach in contrast to an integration analytical style approach.

What we tried to measure was what distance a 5000-kg elephant on roller skates (with an initial speed of 25-m/s rolling down a hill) and with a 1500-kg rocket strapped to its back would have after the rocket finishes a constant 8000-N thrust in the opposite direction with a fuel-burn rate of 20 kg/s.  To solve this problem we calculated the solution with two approaches.  The first approach was to use Newton's second law of gravity (F=ma) to find a function for acceleration (a=F/m) of the elephant with the rocket in respect to time, and then integrate it twice to get an equation for the position of the elephant for at any time (t).  The other approach was to use Microsoft Excel to calculate the distance at any time by entering specific formulas into the needed Excel cells.  Both approaches are based on physics and require mathematics to solve.

This experiment was completed by first looking at the analytical approach that was completed for us in the lab handout.  The integration began with the function of the elephants acceleration, and then the integration was done twice to get an equation for the position of the elephant at any time (t).  The fact that this was already done for us made this part very easy.  Then, the next part required us to use Microsoft Excel so that the program would do automatic calculations with the given parameters and provide us almost instantly with the distance when the elephant's velocity was equal to zero.  Therefore, we entered time, avg acceleration, acceleration, delta velocity, velocity, displacement, and finally distance. Lastly, we entered different delta time values of 0.05 s and 0.1 s to see how it impacted the results.

This is an image of the lab manual that illustrates the integration and the mathematical process that was used to solve for the position equation of the elephant for any time (t).

This image illustrates the final calculation for the elephant of 248.7-m.

This is an image of the Excel graph that proves in a numerical sense that when velocity equals zero, the distance is 248.6 meters.

This image illustrates the values at every 1/10 of a second.  Notice that the distance at 1.5 seconds is the same compared for every second.

This image is for every .5/100 th of a second.  Notice again, that the distance at 1.5 seconds also approximately matches the other time intervals of 36.1 meters.

This image of the work done by my lab partner Kevin Tran illustrates the calculation when some coefficients for the variables changed.  The changes: Initial mass= 5500kg; Fuel burn rate=40kg/s; Thrust force=13000N.  Note: the distance was less.

In conclusion the distance that the elephant traveled due to the rocket's thrust was only 1/10th of a meter in difference.  The analytical approach shows a distance of 248.7m in comparison to the 248.6m that the numerical approach yielded as seen in the prior images.  This is mainly due because the picture I captured for the time interval of every 5/100th of a second was not of the sections where the velocity was equal to zero.  If I would of done that, then I think the distances would of been more precise to one another.  Although, there was a sense that the numerical approach will be faster to calculate when the given variables that were given change, of possibly even if new variables are introduced.  Note that the last image shows a result for a change in magnitudes of some variables and that the distance is less.  Sadly, we did not capture an image of the distance with the new changes so we don't know if that answer matched an answer using the numerical approach.  Also, I feel like there is an equal change to make a mistake in both calculations.  For example, the analytical approach makes a mathematical error easy, and the numerical approach lends to errors during the entering formulas part of the lab.  Consequently, if the numerical approach is entered correctly I feel like it would be a much easier and accurate technique to use.