4-Mar-2015. Non-constant Acceleration Problem.

Problem: 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)= 1500kg -20 kg/s*t.

Find how far the elephant goes before coming to rest.

While using Newton's second law to find acceleration, and then using an equation for the ∆v from 0 to t, v(t) = v + (the integral from 0 to t) a(t)dt, and then deriving the equation once again to find the ∆x give an accurate solution, we found a much easier solution to this problem using excel.

Rather than deriving a complex problem, we used excel to produce the same results by just using the acceleration equation:

a = F/m = -8000 / (650-20t) = -400 / (325-t)

We realized a new way to answer our question, which was by finding the Reimann Sum using excel. The Reimann Sum is an approximation of the area of a region, underneath a curve. The sum is calculated by dividing the region up into shapes, rectangles, to form a region that is similar to the region being measured.

Visualization of Reimann Sum

We then calculated the area for each of these shapes and added them up to get an idea of the larger area, or essentially the integral of a function. We used excel to calculate the many increments needed in finding the most accurate integral.

Beginning of Data Table

The Data Explained:
t: The time gave a time for each acceleration and velocity and position across the data table.
a: This column was formed by using the equation after plugging each time from the time column.
a_avg: This column was found by taking the current row of acceleration plus the previous row and dividing by two to create the length of the rectangle.
∆v: The change in velocity was found by taking the a_avg (length of the rectangle) multiplied by the time increment of 0.05s (width of the rectangle) using the equation ∆v=a-avg*t. The area was therefore considered to be an estimate of the change in velocity (integral of acceleration).
v: The velocity column was found by taking the sum of the original velocity and then each row with the next one from ∆v.
v-avg: This column was found by taking the current row of velocity plus the previous row and dividing by two to create the length of the rectangle once again for another integral approximation.
∆x: The change in velocity was found by taking the v-avg (length of the rectangle) multiplied by the time increment of 0.05s (width of the rectangle) using the equation ∆x=v-avg*t. The area was therefore considered to be an estimate of the change in position (integral of velocity).
x: The position column was found by taking the sum of each row with the next one from ∆x.

We then chose a "small enough" set of ∆t so that when our distance results became more precise. This is when we found that after using the time increment of 0.05 seconds, we got our closest results to the actual solutions of the elephant traveling a distance of 248.7 cm until going into the reverse direction.

The farthest position is boxed in blue on the right. 

In conclusion, we found that same results using excel rather than taking extremely difficult integrals, which was that the elephant travels 248.7cm before reversing direction.