# Clarification on finding acceleration

1. Sep 25, 2014

### Ki-nana18

1. The problem statement, all variables and given/known data
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.)

2. Relevant equations

3. 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.

2. Sep 25, 2014

### 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.

3. Sep 26, 2014

### vela

Staff Emeritus
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 $x$.