02-Mar-2015: Non-Constant acceleration problem solved on excel

Purpose: We will understand the convenience of how excel can calculate solutions from difficult and tedious problems.

We were given a scenario. A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a rocket mounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the
m(t) = 1500 kg - 20 kg/s*t

Find how far the elephant goes before coming to rest.

By hand

Above image: Drawing a picture enabled us to better view the problem. Gathering the given information, we rearranged our force equals mass multiplied by acceleration. Solving for acceleration we came up with our acceleration function of time "a(t)". With a(t) we integrated the equation so that we can get our velocity function of time "v(t)". 

 Above image: Integrating again with the v(t) became even longer because given the way the function was we had to integrate by parts. Doing so eventually lead to our position function of time "x(t)".

With excel

Above Image: We made eight columns; time, acceleration, acceleration average, change in velocity, velocity, average velocity, change in position, and position. Utilizing the functions of excel we where able to essential set up a calculator. B1 was our time interval we set to have excel calculate our values we are searching for. Row 3 is the initial of time, acceleration, velocity and position. B3 we imputed our a(t) we figured out.


The idea of excel is to get the boxes small enough and find a better value.

Left Image: If we image a graph of A vs. T and change in time is small enough. We can approximate the curve as a line. This will give us a trapezoid for a nice formula in finding our velocity. The area underneath the curve will give us the change in velocity. The formula for change in velocity is a result of average acceleration multiplied by change in time. Imagining a graph for V vs. T will similarly follow our last imagined graph. Underneath the V vs. T curve gives us the change in position between the interval.

Right Image: Without integrating, (highlighted in red) excel gave us our answer for the elephant's position before coming to rest. 248.69m

Conclusion: Comparing the way to solve the elephant scenario by hand and using excel is vastly different. Allowing excel to calculate the solution ultimately will be faster and just as accurate if not more. Choosing a small time interval will yield a closer result to the solution by the hundredth decimal place. As you can see from the image above rows 197 to 200 do not change at the hundredths decimal place. Thus finding the velocity as it reaches zero or close to (0.118884) tells us when the elephant is coming to rest. If you come across numbers that do not make sense it could be an error in applying the correct value in excel. A way to tell is if you are given negative signs for a scenario that does not apply.
The logic behind using excel for calculations is to simplify the work load. Typing initial values into the spread sheet is broadly simpler than going through the integration by hand. Ultimately the accomplishments of allowing a computer program to do your calculations will save more time and importantly give you more time to do more.