8-Sept-2014: Modeling non-constant acceleration

The purpose of this lab was to calculate a kinematic motion problem using a non-constant acceleration.

We were first presented with the problem below:

"A 5000-kg elephant on friction less 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) = 1500 kg – 20 kg/s·t.
Find how far the elephant goes before coming to rest."

To begin the problem we had to remember Newton's 2nd Law which states that a force, or sum of multiple forces, is equal to mass multiplied by acceleration.

Fnet = mass x acceleration

We are given both the mass of the system and the force exerted which allows us to manipulate the equation to our needs and convenience in order to solve for the unknown.

acceleration = Fnet / mass

We are now able to calculate acceleration given m(t) = 1500 kg – 20 kg/s·t for the value of the mass and 8000 N giving us the equation:

(-8000 N)/(6500 kg - 20 kg/s t)

The force of the thrust of the rocket  would be negative because we set the positive x motion in the direction of the elephants journey. Simplifying the equation above would give us:

a(t) = (-400)/(325 - t) (m/s^2)

This would be our value of acceleration with respect to time, meaning the value of acceleration of the system varies on the interval of time given. In order to find how far the elephant travels before coming to rest we need to do some simple calculus. By integrating the  value of acceleration we're able to find an equation. The first integral would be as follows:

v(t) = vi + ∫ a(t) 

v(t) = 25 - 400 ln(325) + 400 ln(325-t)

The second integral would give us the value of position and would be as follows:

x(t) = xi + ∫ v(t) 

x(t)=25t-400 ln(325)t+400[tln(325-t)-t-325ln(325-t)+325ln(325)]

Of course this equation is much too complex if we want to solve for t, so in order to find the value of t the equation must be put into Microsoft Excel.