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.

 

[PJC01]

I would suggest the following steps to solve the first order ODE and find x(t):

1. Start by writing the given force equation in terms of acceleration (a) instead of velocity (v). This can be done by using the formula F = ma, where F is the force, m is the mass, and a is the acceleration.

2. Substitute the given force equation into the formula F = ma, and rearrange the equation to get a in terms of v: a = bv^2/m.

3. Now, we have the acceleration equation in terms of v. We can use this to write the first order ODE as dv/dt = bv^2/m.

4. To solve the ODE, we can use the method of separation of variables, where we separate the variables on each side of the equation and integrate both sides. This will give us an equation in terms of v and t.

5. After integrating, we will have an equation in the form of ln(v) = -bt/m + C, where C is a constant of integration.

6. To find the value of the constant C, we can use the initial conditions given in the problem. In this case, the initial position is zero, so we can set v = vi (initial speed) and t = 0 in the equation obtained in step 5. This will give us C = ln(vi).

7. Now, we have the complete equation in terms of v and t, which is ln(v) = -bt/m + ln(vi). We can simplify this to v = vi*e^(-bt/m).

8. To find x(t), we can use the fact that v = dx/dt and integrate both sides with respect to t. This will give us x(t) = -m/(bt) + vi*t + D, where D is a constant of integration.

9. To find the value of D, we can use the initial conditions again. In this case, the initial position is zero, so we can set x = 0 and t = 0 in the equation obtained in step 8. This will give us D = 0.

10. Therefore, the final equation for x(t) is x(t) = -m/(bt) + vi*t.

I would also suggest checking your work and units to ensure that the final equation makes sense and is consistent with the given information. I hope this helps