An athlete with mass m running at speed v grabs a light rope that hangs from a ceiling of height H and swings to a maximum height of h1. In another room with a lower ceiling of height H/2 , a second athlete with mass 2m running at the same speed v grabs a light rope hanging from the ceiling and swings to a maximum height of h2 . How does the maximum height reached by the two athletes compare, and why

It is assumed that each athlete begin at a height of zero. The physical model of each athlete is derived of application of the Principle of Energy Conservation:

Athlete A:

[tex]\frac{1}{2}\cdot m \cdot v^{2} = m \cdot g \cdot h_{1}\\\frac{1}{2}\cdot v^{2} = g \cdot h_{1}\\h_{1} = \frac{1}{2}\cdot \frac{v^{2}}{g}[/tex]

Athlete B:

[tex]\frac{1}{2}\cdot (2\cdot m) \cdot v^{2} = (2\cdot m) \cdot g \cdot h_{2}\\\frac{1}{2}\cdot v^{2} = g \cdot h_{2}\\h_{2} = \frac{1}{2}\cdot \frac{v^{2}}{g}[/tex]

Both athletes reach the same height, as maximum height is a function independent of the mass, but dependent on initial speed and gravity constant.