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

gm 1 e ( c / m ) t c

Result:

Numerical solution of

Time (sec)

Velocity (m/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8.68 15.44 20.70 24.80 27.99 30.48 32.42 33.93 35.10 36.02 36.73 37.28 37.72 38.05 38.32 38.52 38.68 38.81 38.90 38.98

dv c g v dt m

dv v v(ti 1 ) v(ti ) dt t (ti 1 ti ) v(ti 1 ) v(ti ) c g v(ti ) (ti 1 ti ) m c v(ti 1 ) v(ti ) g v(ti ) (ti 1 ti ) m

New Value v (ti 1 ) =Old value +Slope x Step size

Time (sec)

Velocity (m/s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

9.80 17.15 22.66 26.80 29.89 32.22 33.97 35.27 36.25 36.99 37.54 37.95 38.26 38.50 38.67 38.80 38.90 38.97 39.03 39.07