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.