March 08, 2017 Lab 3 Non-constant Acceleration

Lab 3 Non-constant Acceleration

Chris Ceron, Amy Chung, John Choi

Purpose

To calculate the final position of an elephant traveling with a non-constant acceleration using a numerical approach (Microsoft Excel) as shown below.

Theory/Introduction

An elephant on frictionless roller skates reaches the bottom of a hill and arrives on level ground with an initial velocity of 25 m/s. At that point, a rocket mounted on the elephant's back generates a constant 8000N thrust in the Elephant's opposite direction of motion. The mass of the rocket changes with time due to the burning of fuel. 

Sketch of problem

Analytical Approach (Given):

The analytical approach begins with modelling an equation for acceleration as a function of time. Then we take the integral of acceleration to find a velocity function, and integrate once more to find a position function. We then use the function to solve for the time at which the velocity of the elephant is zero, and plug the value into our position function to find the distance traveled in that time.

Numerical Approach:

Using Microsoft Excel, we were able to create functions that calculate when the elephant's velocity is zero, and the position at that time. Through the analytical approach, we know that our time is 19.69075 seconds and the distance the elephant traveled was 248.7 meters. 

Data Breakdown:

Rows:

1 - Initial mass of both the elephant and the rocket

2 - Initial velocity of the elephant

3 - Fuel burn rate

4 - Thrust of the rocket, measured in Newtons

5 - The change of time within our time column

Columns:

A - Time

B - Acceleration

C - Average acceleration

D - The change of velocity

E - Velocity

F - Average Velocity

G - The change in position

H - Position

Data

Data using time interval of 1 second

Data using time interval of .1 second

Data using time interval of .05 second

Conclusion

In this data, we search for when the velocity of the elephant (column E) goes from positive to negative. At some point in between those rows, the velocity of the elephant reaches zero. As we reduce our time intervals, the point of the velocity going to zero gets closer, and our approach in finding the distance becomes more accurate. With a small enough interval, we could find the exact distance the elephant travels. 

1. Through our numerical approach, we were able to accurately find the position of the elephant. The position value found through the analytical approach was 248.7 meters. Through our data, we found the position value to be 248.6966±0.0005m.


2. As our time intervals got smaller, we could see the position of the elephant to change only in the thousandths place. To continue to get more accurate results, you could continue to decrease the time intervals. 


3. Determine how far the elephant would go if its initial mass were 5500kg, the fuel burn rate is 40 kg/s, and the thrust force is 13000 N.

We adjusted our parameters and found that the distance the elephant now travels is 164.03356±0.0009m.