behind the mahem
The Derivation of Our Rocket Equation
According to prior physics knowledge and intuition, we have formulated a picture of the trajectory which we are attempting to model. It can be seen below:
The following is our interpretation of the first two phases of this graph.
In the first phase of its motion, the rocket is acted on by three external forces, gravity, thrust, and air resistance; in our first attempts at producing an equation for the motion of the rocket, we neglected air resistance in hopes that it would turn out to be negligible in the end. After solving the equation for velocity as a function of time and seeing the behavior of the velocity it predicted, we decided that without air resistance, no model of the rockets behavior would be reasonable. We then revised our model to include the effects of air resistance; this model showed the exact behavior that intuition told us to look for. For the details and calculations for both models, take a look at the PDF for phase 1 linked below.
phase1 method and calculations PDF
phase1 method and calculation PDF(version 2)
To illustrate the sharp difference between our two models for the velocity of the rocket during the first phase, here are plots of both the models of phase one
This is the model of velocity as a function of time without taking air resistance into account.
This is the numerically solved velocity as a function of time; notice that the two plots are very similar when the rocket is first starting to gather speed, but as t gets closer to tau the two plots diverge. The first plot continues to increase at a growing rate while the second begins to level off as an effect of the air resistance.
In phase 2, there are only two forces acting on the rocket, gravity and air resistance. We were thefore able to solve for the velocity as an actual function, and with the appropriate initial conditions yielded the graph below.
One detail that indicates that this model is correct is that as the rockets velocity approaches zero, the graph begins to look linear. this is exactly what we would expect from our intuition; when the velocity is small, air resistance is negligible, and the behavior of the rocket is dominated by gravity which is constant. Again the details of our calculations are found inthe PDF linked below
phase2 method and calculations PDF
Java code for numeric integration and ode solving (note: Larry wrote this java code as a supplement to solve our differential equation. It is not yet perfect, but has been of much use. Feel free to browse around.)
Conclusion: after finding the velocity as a function of time, we performed numerical integration to find the maximum height of the rocket is 157 m. This, as well as the velocity of the rocket at its apogee is much larger than we anticipated and than what seems reasonable; we suspect that the value for the coefficent of drag we used may have been too small. We believe a drag coefficent of about 1 would be more likely.