Classical Mechanics - Box sliding down a slope

[jinksys]

I'm on pg 56 of Thorton's Classical Dynamics book and I see this: Imgur Link

Two questions: 1) Where does the 2 go on the second to last equation. 2) Why v0^2 and not v0 on the integral?

 

[fluidistic]

I'm on pg 56 of Thorton's Classical Dynamics book and I see this: Imgur Link

Two questions: 1) Where does the 2 go on the second to last equation. 2) Why v0^2 and not v0 on the integral?

1)$\frac{d}{dt}(\dot x ^2 )=2\dot x \ddot x$.
2)I'm guessing it's because of the differential which is a differential of velocity squared.

 

[richukuttan]

1)$\frac{d}{dt}(\dot x ^2 )=2\dot x \ddot x$.

This is bcoz
$\frac{d}{dt}(\dot x ^2 )=\frac{d}{d\dot x}(\dot x ^2 )*\frac{d}{dt}(\dot x)$
So,$\frac{d}{dt}(\dot x ^2 )=2(\dot x)(\ddot x)$

 

[richukuttan]

2) Why v0^2 and not v0 on the integral?

This is bcoz u r integrating ${d}(\dot x²)$ and not ${d}(\dot x)$.
So, the limits are 0 and v₀².