What is the reason for the division by two in the angular momentum equation?

[Pollywoggy]

Homework Statement

I found this in Goldstein, Poole, and Safko and have seen it in other books. What I don't understand is how the equation gets from the second expression to the third; specifically, why is the m divided by two in the last expression? I am at a loss on this but I know it is not a typo.

Homework Equations

$$\int \mathbf{F} \cdot d\mathbf{s} = m \int \frac{d\mathbf{v}}{dt} \cdot \mathbf{v} dt = \frac{m}{2} \int \frac{d}{dt}(v^2)dt$$

 

[gneill]

$$\frac{d \; v^2}{dt} = 2 v \frac{d v}{dt}$$
so to make this simply $\frac{dv}{dt} v$ it needs to be divided by two.

 

[Pollywoggy]

Thanks, I knew it was something simple that I just could not see.