Clarification on finding acceleration

[Ki-nana18]

Homework Statement

If a block slides w/o friction down a fixed, inclined plane with an angle theta of 30 degrees, what is the block's acceleration? ​

(This problem is an example from my textbook, meaning the solution is available. I am confused on the method that they used. Any clarification is greatly appreciated.)​

Homework Equations

The Attempt at a Solution

I understand the process on how to determine the acceleration, I also understand the that to find velocity you integrate acceleration, I don't understand the purpose of the multiplication by 2(overdot)x. It seems as though its used to manipulate the constant acceleration.[/B]

 

[BiGyElLoWhAt]

It let's you set up a derivative product rule on the left hand side. It's a similar method to an integration factor when solving linear first order differential equations. This is the first time I've seen this done, though. Seems legit.

 

[vela]

You're starting with $\ddot{x} = \frac{dv}{dt} = g\sin\theta$. When you integrate, you'd get
$$\int_{v_0}^v dv = \int_{t_0}^t g\sin\theta\,dt = (g\sin\theta)(t-t_0),$$ which is fine if you're trying to calculate how fast the block is moving after a certain time. The problem, however, is asking you to find the speed after the block has slid a distance $x_0$. By introducing the factor of $\dot{x}$ on the righthand side, you're changing variables from $t$ to $x$.