Energy conservation of a child on a swing

[mikefitz]

The maximum speed of a child on a swing is 4.9 m/s. The child’s height above the ground at the lowest point is 0.7 m. How high above the ground is he at his highest point?
http://img208.imageshack.us/img208/9949/untitledyx0.png

Above is the work done to solve the problem. I don't understand how they went from the blue to the red equation. Wouldn't the two g's cancel out since you would need to divide the left side by mg?

 

[Doc Al]

Wouldn't the two g's cancel out since you would need to divide the left side by mg?

You divide all terms by mg. The m's all cancel, since an m appears in all three terms. But g's only cancel in the two terms that had g's--the KE term does not have a g to cancel.

 

[m2003]

The blue equation represents the kinetic energy of the child at the highest point, which is equal to the potential energy at the lowest point. The red equation is derived from the conservation of energy principle, which states that the total energy of a system remains constant. In this case, the total energy is the sum of kinetic and potential energy, which is equal at both the highest and lowest points. Therefore, we can equate the two equations and solve for the height at the highest point. The g's do not cancel out because they represent different quantities - acceleration due to gravity (g) and the height (h).