Can anyone explain this integration for me?

[Rijad Hadzic]

Homework Statement

http://imgur.com/a/Y8NW0

Basically we start with a function of t, which was differentiated twice, that function = $F_o / m$

Fo is a constant force, and I assume m is mass though my book doesn't state that.

Homework Equations

The Attempt at a Solution

Integrating the function, I multiply m to the other side, because I assume m is mass and mass is a constant. so it shouldn't really matter.

since F_o is a force, that means it is Kg * m/s^2, divided by mass which is in kg, you get m/s^2 which is indeed acceleration, the second derivative of time.

Integrating, I get

$m\int x^{''}(t) \, dt = \int F_o \, dt$

$m\int x^{''}(t) \, dt = F_o(t)$

But as you see from the link I provided, the book states
$m\int x^{''}(t) \, dt = F_o(t) + V_o$

First off, I don't even know what V_o is. Is it inital velocity, or is it final velocity? It would make sense for it to be final velocity since $m\int x^{''}(t) \, dt$ is literally velocity itself..

I was under the impression that if you integrate something with respect to time, and that function does not have a time variable, it would simply be that function times the variable t.

Why is it that they are adding V_o in this example?

 

[Rijad Hadzic]

I marked it as solved. I guess I didn't read clearly enough, V_o is the constant of integration.

That makes sense now. Always end your integrals with + C, I feel dumb now..