Solve the differential equation F=F0+kv

[OmegaKV]

Homework Statement

Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force:

$$F=F_0+kv$$

You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v, not t. The answer in the back of the book involves ln.

Homework Equations

Maybe this:

$$\ddot {x}= \frac{d \dot{x}}{dx}$$

The Attempt at a Solution

$$m\ddot{x}=F_0 +k\dot{x}$$
$$m\dot{x} \frac{d\dot{x}}{dx}=F_0 +k\dot{x}$$
$$m\dot{x} d\dot{x} = F_0dx + k\dot{x} dx$$
$$\frac{1}{2}m\dot{x}^2 = F_0*x + ?$$

 

[Orodruin]

The second equation you wrote down in your attempted solution is separable, but in order to separate it the side with dx should not depend on $\dot x$.

 

[rude man]

Homework Statement

Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force:
$$F=F_0+kv$$
You could say mv = F0*t + kx, but the answer in the back of the book is an equation that is only in terms of x and v, not t. The answer in the back of the book involves ln.

Homework Equations

Maybe this:
$$\ddot {x}= \frac{d \dot{x}}{dx}$$

Is this eq'n dimensionally correct?

The Attempt at a Solution

$$m\ddot{x}=F_0 +k\dot{x}$$
$$m\dot{x} \frac{d\dot{x}}{dx}=F_0 +k\dot{x}$$
$$m\dot{x} d\dot{x} = F_0dx + k\dot{x} dx$$
$$\frac{1}{2}m\dot{x}^2 = F_0*x + ?$$

 

[Orodruin]

Is this eq'n dimensionally correct?

Well, he is not actually using that equation. He is using $\ddot x = \dot x \, d\dot x/dx$.

 

[rude man]

Homework Statement

Find the velocity of v as a function of displacement x for a particle of mass m which starts from rest at x=0 and subject to the following force:
$$F=F_0+kv$$

The Attempt at a Solution

$$m\ddot{x}=F_0 +k\dot{x}$$

Assuming F₀ and k are constants, how about a substitution of variables to reduce the 2nd order linear ODE into a 1st, then taking orodruin's hint to employ separation of variables to solve the new equation?

 

[rude man]

Well, he is not actually using that equation. He is using $\ddot x = \dot x \, d\dot x/dx$.

Right, but he wrote it down & should learn to check for dimensional consistency, a powerful error-detecting tool that doesn't seem to be sufficiently emphasized in our classrooms.