Elephant jet pack!

Purpose: The activity is designed to have the students look at a problem that has a non-constant acceleration and solve it using two methods, one is with calculus and the other is using excel to do hundreds of calculations to find the desired value.

Equipment: No equipment used except for this activity.

Initial problem:
The actual problem is that there is an elephant with a rocket on its back, the elephant has a mass of 5000kg and the rocket has a mass o 1500kg. The elephant is on frictionless roller skates. The elephant goes down a hill and when it reaches the flat horizontal ground it has an initial velocity of 25m/s. That is considered t=0 and at t=0 the rocket ignites and produces a constant force of 8000N in the opposite direction of the initial velocity. As the rocket burns, it burns 20kg of fuel at every second. At what position would the velocity be equal to zero?

Solution:
Method 1) Newton's Second Law of Motion states that Force=Mass*Acceleration, so there is a function for mass and a constant force so the expression for acceleration would be the force divided by the mass, which is changing.
This is completely solvable using calculus. The function of acceleration is simply the second derivative of position, so to get to the position function it requires integrating twice, also including the initial conditions for velocity before the integration of the velocity function.

The t when velocity is zero is needed. At v=0m/s, t=(19.68s) and t at v=0m/s is plugged into the position function.
Method 2) Instead of doing the tedious calculus, numerical integration can be done in Excel.

The first column is time, the second column is the acceleration, the third column is the average acceleration in the time interval, the fourth column is the change in velocity during that interval, the fifth interval is the actual speed at the specific time, the sixth column is the change in position during the time interval, and the final column is the actual position the elephant is relative to its initial point at the bottom of the hill. The first column, time, requires no calculations and is the determining factor. The smaller that the values increases by, the more accurate the other values become. The smallest value that time was set for was .05s. The second column is simply a matter of plugging in time to -400/(325-t) and finding the acceleration that that point in time. The third column is an average between the two accelerations. The fourth column calculates the change in velocity by multiply the average acceleration by the time interval. The fifth column is taking the initial velocity, which is 25m/s and adding the change in velocity calculated in the fourth column, it should be noted that it is deceleration so the change in velocity is always zero until the rocket stops and the elephant reverses direction. The sixth column is the change in position which is calculated by multiply the average velocity in the time interval by the time interval. The final column is the position of the elephant which is calculated by adding the change in position to the initial position, which is 0.

Conclusion:
The results from the Excel calculations were very accurate. The analytically calculated value for the position of the elephant when the elephant stops completely was 248.697m and the value that the numerical method resulted was 248.698m. So the error was less less than a centimeter.

Sincerely,

Swaggy C