You have modelled an ODE for the velocity of your car. You have made two models,

Question

You have modelled an ODE for the velocity of your car. You have made two models, one where air resistance is proportional to velocity (less accurate but easier to solve), and one where air resistance is proportional to velocity squared (more accurate model). The ODEs derived from this process are given below.

Simple ODE: dv/dt + c1/m*v = F(t)/m

Accurate ODE: dv/dt + c2/m*v^2 = F(t)/m

You want to model the velocity of your car over time. The mass of the car is 1000 kg, and F(t) is a constant force of 10000 N. The initial velocity is 0, while the top speed is 90m/s (there is no acceleration once top speed is reached).

1. Solve the damping coefficient for the simple model, c1 , and the accurate model, c2 . Note that you do not need to solve the ODEs to determine these values.

Explanation / Answer

Solution :-    Given Simple ODE : dv/dt + C1/m*v = F(t)/m

m=1000 kg , F(t) = 10000 N , v0= 0 , at last Final velocity vf= 90 m/s

As already mentioned that at velocity vf acceleration (dvf /dt) = 0

so using this to solve Simple ODE for value of c1

dv/dt + C1/m*v = F(t)/m

dvf/dt + C1/m*vf = F(t)/m as dvf/dt =0   

0 + C1/(1000*90) = 10000/1000

C1/90000 =10

Ans :-   So Value of C1= 900000 or C1 = 9* 105

Similarly as above for Accurate ODE dv/dt + C2/m*v2 = F(t)/m

m=1000 kg , F(t) = 10000 N , v0= 0 , at last Final velocity vf= 90 m/s

As already mentioned that at velocity vf acceleration (dvf /dt) = 0

so using this to solve Accurate ODE for value of c2

dv/dt + C2/m*v2 = F(t)/m

dvf/dt + C2/m*vf2 = F(t)/m    as dvf/dt =0     

0 + C2/(1000*902) = 10000/1000

C2/8100000 =10

Ans :-   So Value of C2= 81000000 or C2 = 8.1*107

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