Purpose: The purpose of the lab is to solve an equation with a non-constant acceleration mathematically by using an excel spreadsheet.
Problem: There is a 5000 kg elephant on skates who is on a hill which is frictionless, as he comes to the bottom of the hill. The rocket is then thrusted on and lets out a force of 8000N(newtons) constantly, but it is pointed in the opposite direction of where the elephant is moving. The mas of the rocket is its mass before the rocket is on, 1500kg, minus the fuel which leaves the rocket causing it to lose fuel at 20kgs a second. Meaning mathematically the rockets mass is 1500-20kg/s*t=M(rocket).
The question is asking us to find the distance the elephant has traveled before running out of fuel or coming to rest.
Procedure: In order to solve for the equation we worked it out mathematically and we also used excel to start a spreadsheet in order to input different data and test our results to find a time to then find a distance that the elephant has traveled. Mathematically speaking we labeled all of our known values.
M(rocket)=1500kg-20t
elphant mass=5000kg
v(intitial)= 25m/s
masses together= 5000+1500-20t=6500-20t
We can use the equation of the masses to solve for a force value which gives us our a(t), acceleration function.
The picture above shows the integral of a(t)= Fnet/M(together) which equals F(t)= -8000/6500-20t(all functions integrated from 0 to t) a(t)= -400/325-t. Leading to the equation v(t)=25-400ln(325)+400ln(325-t). After taking the integral of this function we find the position equation x(t)= (25-400ln(325))t+400(t-325)*ln(325-t)-t+325ln(325), which we need to find the time he traveled in order to find the distance that the rocket and elephant traveled. We found that time was equal to 19.69 seconds by solving for velocity at V(0). We use this t value to then solve for the distance the elephant had traveled which approximated to about 248.7. To prove our mathematical work we used excel spreadsheet to calculate the a(t),v(t), and x(t) by setting up several different columns where we input equations with respect to what they represent.
Each column was labeled,( t) represents time, (a) represents acceleration, (a-avg) represents the average acceleration on a given time interval, (deltaV) change in velocity on a given time interval,(v) represents velocity, (Va) represents the average velocity on a time interval, (deltaX) represents the distance traveled in a time interval, (X) represents the total distance traveled completely. We used a small time interval to guarantee accuracy. A was represented by the acceleration equation we integrated. A-Average was found by a(final)-a(initial)/2. deltaVelocity = acc*time.Velocity = V(initial)+ deltaV(change in v).Velocity (average)= Vf-Vi/2. DeltaX= A-avg*t. X is the position which is solved by plugging in each time for x(t) equation above.
Class data: In class we were asked to have a very large time scale in order to check our work so we had the columns go down an addition two hundred rows. By simply highlighting the row and the equation used to solve for the respective value we got around 230 new columns with new data. We continued to do so with every row until our entire data had each column down to about 230 rows each.
The second way of solving is by going to the velocity column and checking where time is equal to zero because we want to know the distance that the rocket and elephant traveled after coming to rest when the rocket runs out of fuel. We find that our data matches!!!! The excel sheet calculated a 19.6-19.8 time window since we made our time interval so small were were able to see just what a tenth of a second would change. Our time we found was 19.69 seconds approximately which is between the time window we found on excel.
Final Thoughts: Sometimes when solving numerically and analytically the resultant is equal making your working easy knowing that you did the correct steps mathematically to solve for the problem in this case a non-constant acceleration problem involving an elephant on skates thrusted by a rocket strapped to his back. The elephant traveled a distance of 248.7 m before coming to rest which we proved in both methods, numerically and analytically. The thing that allowed us to get such accurate data was the fact that our time interval was shortened to a tenth of a second allowing more data and more accurate numbers to be calculated. This lab helped to retouch on skills on excel to analytically solve something by first proving it mathematically.
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