Proving a complex Force derivation with CalcConfused

[BreakPoint]

Homework Statement

If we know F(t) the force as a function of time for straight line motion, Newton's second law gives us a(t) the acceleration as a function of time. We can then integrate a(t) to find v(t) and x(t). However, suppose we know F(v) instead. a) The net force on a body moving along the x-axis equals -Cv^{2}. Use Newton's second law written as ƩF = m \stackrel{dv}{dt} and two integrations to show that x - x0 = (m/C) ln(v0/v). b) Show that Newton's second law can be written as ƩF = m*v*\stackrel{dv}{dx}. Derive the same expression in part (a) using this form of the second law and one integration.

Homework Equations

F = ma
F(v) = -Cv^2

The Attempt at a Solution

For the first part a), I had no idea so I started to differentiate the answer to try and get a method for simplifying it originally. The problem I encounter is that at X, If I take the 2nd derivative of the function I get a(t) = m/Cv^2, which seems impossible because you can't do that with -Cv^2 = m dv/dt. I solved that for dv/dt but i get nowhere really. I'm extremely stumped. Any pointers in the right direction would be extremely useful.

Thanks in advance.

 

[Tsunoyukami]

For part a, you already know that

F = ma
F(v) = -Cv^{2}

We can then write the following:

ma = -Cv^{2}
a = \frac{-Cv^2}{m}

\frac{dv}{dt} = \frac{-Cv^2}{m}


Next I will express \frac{-C}{m} as some other constant to simplify the algebra. So let \alpha = \frac{-C}{m}.

\frac{dv}{dt} = \alpha v^2

We can then use the method of separation of variables from ordinary differential equations to write:

\frac{dv}{v^2} = \alpha dt

\int \frac{dv}{v^2} = \alpha \int dt

You should be able to solve this integral for an expression for v. Then, if you isolate v you will be able to write v = dx/dt and repeat this procedure to find an expression for x.