Theory: That if the time integral is small enough while using excel, we can determine how far the hypothetical 5000kg elephant, on frictionless roller skates, going 25m/s at first, then a rocket on its back generates 8000N thrust opposite its current motion, and the rockets mass changes over time and can be expressed m(t)=1500kg-20kg/s*t, numerically rather than analytically.
Procedure: Inputting formulas into excel
*First we set up the basic acceleration equation (acceleration=force/mass) with time as the only variable, given the situation. The elephant has a -8000N force heading in the opposite original direction and the mass is 5000kg(elephant) plus a rocket of 1500kg which is losing mass at 20kg a second.
Acceleration=-8000N/(6500kg-20kg*Time) or reduced too a= -400/(325-t)
*We input this formula into one of the cells, scroll down a satisfactory length, here about 220 rows.
*For time we're going to do increments of only .1 seconds, so we create a column that constantly increases by only .1 seconds, scroll down 220 rows, and have that time be placed into the adjacent cell for calculating acceleration.
*In another column we can now take the average acceleration of each preceding two calculated accelerations and divide by two.
*If we multiply the average acceleration by the each increment of time (0.1 seconds), we have calculated the change in velocity, which since the elephant is slowing down should be negative. Scroll down to make a column
*Adding the change in velocity (really how much the velocity is decreasing by) to the original results in the velocity at the specific time. Scroll down to make a column.
*With a column of velocity we can create an average velocity at each moment, using the same method as for average acceleration. Scroll down to create a column
*With a column of average velocity we can determine the change in distance by multiplying by the 0.1 second time increment for each cell. Scroll down to create a column
* Adding each increment of change in distance for a total distance travelled.
*If everything was done correctly the first few rows should look like this
Results
|
T s
|
a m/s^2
|
a_avg m/s^2
|
∆v m/s
|
v m/s
|
v_avg m/s
|
∆x m
|
x m
|
|
0
|
-1.23076
|
0
|
25
|
0
|
|||
|
0.1
|
-1.2311
|
-1.23095
|
-0.12309
|
24.87690
|
24.93845
|
2.493845
|
2.493845
|
|
0.2
|
-1.23152
|
-1.23133
|
-0.12313
|
24.75377
|
24.81533
|
2.481533
|
4.975378
|
|
0.3
|
-1.23190
|
-1.23171
|
-0.12317
|
24.63059
|
24.69218
|
2.469218
|
7.444597
|
*If we scroll down, eventually the increments in distance start to decrease instead of decrease, because the average velocity is becoming negative, meaning the elephant is finally moving in the opposite direction around the 19.7 second mark.
Results
|
19.6
|
-1.30975
|
-1.30954
|
-0.13095
|
0.118883
|
0.184360
|
0.018436
|
248.692701
|
|
19.7
|
-1.31018
|
-1.30997
|
-0.13099
|
-0.01211
|
0.053385
|
0.005338
|
248.69804
|
|
19.8
|
-1.3106
|
-1.31040
|
-0.13104
|
-0.14315
|
-0.07763
|
-0.00776
|
248.690276
|
*Looking at the chart this is the furthest the elephant goes 248.7 meters, and solving the original equation for acceleration through integration is also 248.7 meters.
Conclusion
*In comparison, the numerical method is a valid form to determine an answer over the analytical method, as long as you have a calculator that can do all the blunt force calculations. As excel has determined the distance traveled, just as hand written calculus would.
*The smaller the integral of time the smaller the change in calculating the distance at each point in time, too large could result in a few meters changing in each cell vs the 1/100th of change. So the integral would be "small enough" depending on how many significant figures you need.
*You know if you've reached the max point of distance travelled when the distance is starting to decrease, meaning the elephants velocity is finally heading in a different direction.
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