How Do You Solve for Position x(t) Given a Force Dependent on Velocity?

[hanilk2006]

A particle of mass m is subject to a force F(v) = bv^2. The initial position is zero, and the initial speed is vi find x(t)

so far

m*dv/dx*v = -bv^2
m*dv/dx = -bv
integral m/-bv*dv = integral dx
m/-b*ln(v) + a = x + b

What do I do with the constants? i thought i was suppose to put in 'a' as vi and b as 0, but then when i integrate again for v, so i can get x(t) function, what do i use to fill in that constant?

 

[UltrafastPED]

Where does m*dv/dx*v = -bv^2 come from? Newton's second law of motion implies F=m*dv/dt.

 

[hanilk2006]

dv/dx*v=dv/dx*dx/dt = dv/dt = a

 

[UltrafastPED]

OK. You use your constants in the integrations: the lower constant on the dx integral is the starting location = 0, and the lower constant on the dv integral is the initial velocity = v_i.

The upper constants are the unknowns ... your integral equation will provide a relation between them.