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Problem Statement: Parachutist: Two forces act on a parachutist, the attraction by the earth mg (m mass of person plus equipment, the acceleration of gravity g=9.8 m/sec^2) and the air resistance, assumed to be proportional to the square of the velocity v (t). Using Newton’s second law of motion (mass acceleration resultant of the forces), set up a model (an ODE for v (t)). Graph a direction field (choosing m and the constant of proportionality equal to 1). Identify the falling velocity of a falling parachutist by using numerical method. What is the limiting velocity?
Mathematical Model: Newton’s 2nd law of Motion States that “the time rate change of momentum of a body is equal to the resulting force acting on it.”. The model is formulated as F=ma F=net force acting on the body (N) m=mass of the object (kg) a=its acceleration (m/s2) Parachutist:
Figure 1: Schematic diagram of the forces acting on a falling parachutist. F D is the downward force due to gravity. Fu is the upward force due to the resistance
F ma dv F dt m F FD FU FD mg FU cv dv mg cv dt m
dv c g v dt m
This is a differential equation and is written in terms of the differential rate of change dv/dt of the variable that we are interested in predicting.
Analytical solution of
dv c g v dt m
Using linear differential equation:
dv c vg dt m Integration factor: c
I e
m dt
Solve it: (
)
(
) ∫ (
)
At initial condition v(0)=0
so the final solution after simplify its become
v(t ) Let’s defined a constant : Mass of parachutist is 50 kg, Gravitational force is 9.81m/s^2
