Compute the velocity of a freefalling parachutist using Eule

Compute the velocity of a free-falling parachutist using Euler

Solution

By newton\'s second law of motion we have

m dv/dt =- c_dv² +mg => dv/dt =- c_dv²/m+g

Now writing this differential equation as differences, we get:

v(t_i+1)-v(t_i)/(t_i+1-t_i+1) = -10v(t_i)²/80+9.81 = -v(t_i)²/8+9.81

=> v_i+1-v_i = (-1/8v_i²+9.81)(t_i+1-t_i)

Now we are to use step size of 1s so that means t_i+1-t_i = 1

Hence v_i+1 = v_i -1/8v_i²+9.81

Start with t₀ = 0 , v(0) = 20 = v₀

Now t₁ = 1, v₁ = v₀+(1)(9.81-v₀²/8) = 20 + 9.81-50 = -20.19

t₂ = 2, v₂= v₁+(1)(9.81-v₁²/8) = -20.19+9.81-50.9545125 = -61.3345125

Similalrly, t₃ = 3, v₃ = -521.7648

t₄ = 4, v₄ = -34541.77015

t₅ = 5, v₅ = -149176267.5613

t₆ = 6, v_{6 =} -2,781,694,999,615,456.86

t₇ = 7, v₇ = -9.67228 x 10^-29

And at last t₉ = 9, v₉ = -1.709 x 10^-117

Now at t=10 s, the cd = 50kg/s

So our equation now changes to:

dv/dt = -50v²/80+9.81 = -0.625v²+9.81

So v₁₀ = v₉ +(-0.625v₉² +9.81)

And the values are getting very large but procedure is same