Center Mass Acceleration of 2 bodies on different inclines.

[eginon]

[Edit: ignore the below I realized what it represented later this afternoon. Probably because I asked and stopped thinking about it.]

Jack is at the top of a hill that is 35 degrees. Jill is at the bottom of the hill on another slope which is 22 degrees. Determine the acceleration of the center mass. Jack and Jills masses are given but I think they are probably not relevant to the question I have, let's just call them m1 and m2. Based on looking at other questions I know that the solution is

$\frac{1}{M}(m1 g sin(\theta) cos(\theta) + m2 g sin(\phi) cos(\phi))$Where M=m1+m2 , θ represents the angle of Jack's slope, and $\phi$ is the angle of Jill's slope

I get that Jack's acceleration due to gravity is a=g sin(θ). Before I looked up the correct answer I was mistakenly using just a = g sin θ. What does the cos of θ and $\phi$ of each angle represent in relation to center mass?

 

[Nate810]

The cosine of the angles represents the component of the acceleration due to gravity that is in the direction of the center of mass. This is because if you draw a triangle between Jack and Jill, the cosines of the angles will be proportional to the ratio of the distances between Jack and Jill and the center of mass.