Your instructor will explain to you a method of using a Personal Computer to model particle dynamics. The technique, performing repeated iterations of a description of motion, allows us to follow the particle through small changes. Since you can make the computer do a lot of of these small changes quickly, you can follow a particle throughout its flight, as long as the changes do not get too big. The technique is called Euler's Method. Many regard him as the greatest mathematician of all time. He published 886 books and more than 3000 of his letters about mathematical ideas are still around. He is the guy that defined symbols like the summation sign, integral sign, f(x), p and even e (base of the natural logarithm.)
Euler Method:
Recall from definition that
a (t) = Dv / Dt = { v( t + Dt ) - v(t) } / Dt
So we can write the velocity as:
v( t + Dt ) = v(t) + a(t) Dt
Similarly we can write:
v (t) = { x (t + Dt ) - x (t) } / Dt
so that:
x (t + Dt ) = x (t) + v(t) Dt
This is not unlike the usual equations of motion except that the 1/2 a (Dt )2 does not appear. It turns out that for really small Dt , when squared, becomes so small that it is insignificant in the result. For example, when 0.01 is squared the result is .0001.
If we assume that
a (x, v, t) = F (x, v, t) / m
from our usual F = m a in Newton's second law, we basically have a numerical way to solve motion problems. This is so powerful, that flights to the moon and ICBM trajectories are calculated this way, with very very litle error. The results are basically "close enough for gov't work."
You will use your PC and EXCEL spreadsheet to solve the following
problem:
A 50 kg parachutist jumps from an airplane and falls to the Earth with a drag force proportional to the square of the velocity, R = D v2 . Take D = 0.2 kg/m with the parachute closed and D = 20 kg/m with the chute open. Do the following:
(a) Determine the
terminal
velocity of the parachutist in both configurations, before and after
the
chute is opened. You don't need the computer for this. Calculate it by
drawing a free body diagram and sum up forces.
(b) Set up a
numerical
analysis of the motion and compute the velocity and position as
functions
of time. Assume she began her descent at 1000 m above the ground and
fell
in free fall for 10 seconds before opening the chute. (Hint: when the
parachute
is opened, there is a tremendously large acceleration, or deceleration;
a smaller time step during this phase may be necessary.) Basically,
you'll
need three parts, the free fall, the opening, and the flight with the
chute
opened. Assume motion in the vertical direction only. Enjoy this
challenging
exercise.