Question

A 37 kg child jumps in the air. At the maximum height of the jump, the GPE is 500J. If the mass of the boy is tripled, how much Potential Energy does the child have at the top of his jump? If the same boy's mass is doubled, but his height is quadupled, how much Potential Energy does the child have at it's maximum and in what position in the jump will he be?

Answer 1

*Answers are in their respective order with the questions.

`1500\:\mathrm{J},\\4500\:\mathrm{J},\\5.52\:\mathrm{m}`

Step-by-step explanation:

The [gravitational] potential energy of an object is given by
`PE=mgh`, where
`m` is the mass of the child,
`g` is acceleration due to gravity, and
`h` is the height of the object (relative to the ground).

"If the mass of the boy is tripled, how much potential energy does the child have at the top of his jump?"

Without actually plugging and chugging, notice the formula in which gravitational potential energy is given. Potential energy is directly proportional to mass. Therefore, if the mass of the boy is tripled, his potential energy will also be tripled, yielding an answer of
`500\cdot 3=\boxed{1500\:\mathrm{J}}`

"If the same boy's mass is doubled, but his height is quadrupled, how much potential energy does the child have at its maximum and in what position in the jump will he be?"

In similar fashion to the first question, we can note that the potential energy of an object is directly proportional to its mass and height. Therefore, if the boy's mass is doubled and his height is quadrupled, his potential energy will be
`2\cdot 4=8` times as large at maximum height. Thus, he will have
`500\cdot 8 =4,000\:\mathrm{J}` of potential energy at the maximum height. I'm assuming that "in what position in the jump will he be?" is asking for the height at the maximum height of the jump.

The original height can be solved using:

`PE=mgh,\\500=37\cdot 9.8\cdot h,\\h=(500)/(37\cdot 9.8)=1.37892995036\:\mathrm{m}`

Quadrupled implies being multiplied by four. Therefore, we have:

`h'=1.37892995036\cdot 4=5.51571980143\approx \boxed{5.52\:\mathrm{m}}`