Here's the question:
My book says the answer is C. How is it not A?
I know that all the potential energy is transferred to kinetic energy. With algebra, knowing Kinetic energy is (1/2) * m * v^2 and gravitational potential is mgh, I solve for h which results in (v^2)/2g
Ok so since this is a proportional reasoning problem my focus is that h = v^2 meaning height is directly proportional to the square of velocity. That being said, if we half the velocity, that means that some value (let's use p for the variable) multiplied by height gets me to (1/2v)^2.
That being said, if we refer back to the equation h = (v)^2 and v is being halved, it should look like this:
h * p (<- growth factor) = (0.5v)^2. Now it looks like the right side of the equation grew by a factor of (1/4). Think about it. If your velocity is 10, the right side of the equation becomes 100. If you half that velocity, the right side of that equation becomes 25. You can see that the right side of the equation grew by a factor of 1/4. That means that $p$ (my growth factor variable) should also be (1/4). $h * 1/4 = h/4$. NOT $3h/4$. Where did I go wrong?