Question

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 1 h . in another room with a lower ceiling of height 2 h , 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 2 h . how does the maximum height reached by the two athletes compare, and why?

Answer 1

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Remember that the mechanical energy $E_m=E_k+E_p$ of the system, (where $E_k, E_p$ are the kinetic energy and the potential energy respectively) must remain the same. In an initial situation, the athletes have purely kinetic energy. Then they hang on to the rope and reach a maximum height, at which point they have zero speed and thus zero kinetic energy. The athletes now have purely potential energy, remember again that the mechanical energy must be the same than it was initially, this implies that:
$E_k=E_p\implies\frac{1}{2}mv^2=mgh_1\implies h_1=\frac{V^2}{2g}$
For the first athlete, and
$E_k=E_p\implies\frac{1}{2}(2m)v^2=mgh_2\implies h_2=\frac{V^2}{g}$
for the second athlete.
We can see clearly that the second athlete rose higher than the first athlete.