Rock climbing on Mars

In general, the acceleration due to gravity at the surface of a planet ( $g_p$ ) is given by:

$g_p = \frac{GM_p}{R_p^2}$

where $G$ is the gravitational constant, $M_p$ is the mass of the planet, and $R_p$ is the radius of the planet. Thus, the acceleration due to gravity on Mars would be:

$g_M = \frac{GM_M}{R_M^2} = \frac{G(0.11M_E)}{(0.53R_E)^2} = \frac{0.11}{0.53^2} \frac{GM_E}{R_E^2} = 0.39g_E$

where $M$ and $E$ subscripts denote Mars and Earth, respectively. Since the amount of work the rock climber does each time he climbs his route is inversely proportional to the acceleration due to gravity, we get:

$\frac{20}{x} = \frac{g_M}{g_E}$

where $x$ is the number of times he can climb the route on Mars. Solving for $x$ , we get:

$x = 20\frac{g_E}{g_M} = 20\frac{g_E}{0.39g_E} = \frac{20}{0.39} \approx \boxed{50}$

Since the amount of energy Sam has is the same both on Mars and on Earth, the work done by the gravitational forces is going to be the same in both cases. This means that :

$\frac{20md(M_E)G}{R_{E}^2} = \frac{kmd(M_M)G}{R_{M}^2}$

where

$G = \text{Gravitational Constant} \\ M_E = \text{ Mass (Earth)} \\ M_M =\text{ Mass (Mars)} \\ { R }_{ E } = \text{Radius (Earth)} \\ { R }_{ M } = \text{Radius\quad (Mars)}\\ m = \text{Rock climber's mass}\\ d = \text{Mountain Height}\\ k = \text{times he climbs the mountain}$

So, as :

${ M }_{ M }\quad =\quad 0.11*{ M }_{ E }\\ { R }_{ M }\quad =\quad 0.53*{ R }_{ E }\\$

After the caulculations :

$k\quad \cong \quad 50$

The force of gravity that Sam is working against on Earth is proportional to his mass, m, times the mass of Earth, M, divided by the square of the distance between them (i.e. the radius of the Earth, r).

or

$\displaystyle \frac{Mm}{r^2}$

If the mass of Mars is 11% that of Earth and its radius is 53% that of Earth the equation becomes

$\displaystyle \frac{0.11Mm}{(0.53r)^2}=\frac{0.11Mm}{0.28r^2}=0.39\frac{Mm}{r^2}$

So, if Sam can do the climb 20 times on Earth, he should be able to do it $\displaystyle \frac{20}{0.39}= 51$ times on Mars.