**HOW HIGH CAN YOU JUMP ON MARS?**

And while we are at it...

- How high can you jump on the Moon?
- How high can you jump on Jupiter?
- How high can you jump on Ceres?

Suppose you can supply the same kinetic energy to a jump no matter where you are. This would mean your initial (launch) velocity v is the same, no matter what the local gravity is.

If local the local gravity is g, and you leave the ground with velocity v, your peak height h is given by:

If Earth gravity is g_{e}, but your local gravity is g_{x}, then the ratio of peak heights is g_{e}/g_{x}.

For reference,
Earth gravity is 9.807m/s^{2},
Mars gravity is 3.711m/s^{2},
and Moon gravity is 1.620m/s^{2}.

So compared to Earth, you could jump 2.64 times higher on Mars, and 6.05 times higher on the Moon.

This is a widely published answer, with at least 10,000 references on the web,
but **it is completely incorrect!**

Let's think about this again.

If local gravity is low, you weigh less, and so your legs can **push**
your body mass upward faster. If local gravity is high, you weigh more,
and your legs will struggle to **push** your body upward. If the gravity
is sufficiently high, you cannot jump at all.

To push (or pull) is to apply a force. As you jump your legs push up as gravity pulls down. We know your mass (m) is constant, and for simplicity we'll assume the force (F) your legs apply is constant. Force equals mass times acceleration.

When you jump on Earth your leg force is:
_{e}+g_{e})

When you jump on a planet with local gravity g_{x} your leg force is:

F = m (a_{x}+g_{x})

The force F is the same in both cases. Setting the two equal yields:

a_{x} = a_{e} + g_{e} - g_{x}

Force is applied over a distance L, which is limited by your leg length. The launch velocities are therefore:

v_{e}^{2}
= 2 L a_{e}

v_{x}^{2}
= 2 L a_{x}

= 2 L ( a_{e} + g_{e} - g_{x} )

= 2 L a_{e} + 2L ( g_{e} - g_{x} )

= v_{e}^{2} + 2L ( g_{e} - g_{x} )

Suppose the maximum height an athete can reach by jumping on Earth is
h_{e}. We can write an upper bound on v_{e} in terms
of h_{e} and g_{e}:

v_{e}^{2}
≤ 2 · h_{e} · g_{e}

And therefore:

v_{x}^{2}
≤ 2 · h_{e} · g_{e}
+ 2 · L · ( g_{e} - g_{x} )

The maximum height of a jump on a planet with gravity g_{x} is threfore:

h_{x}
≤ [ 2 h_{e} g_{e}
+ 2 L ( g_{e} - g_{x} ) ]
/ (2g_{x})

= h_{e} g_{e} / g_{x}
+ L g_{e} / g_{x}
- L g_{x} / g_{x}

= h_{e}·( g_{e}/g_{x} )
+ L·( g_{e}/g_{x} - 1 )

While most people on Earth can jump to extend their reach perhaps 12 inches, the world record standing vertical jump is a staggering 46 inches (1.16m), set by NFL player Gerald Sensabaugh. The launch velocity for this jump was 10.67mph (4.77m/s).

This leaves L as the only variable. This range is simlar to the distance a weight moves when using a leg press at the gym. Whether a 5'1" gymnast, a 6'1" Sensabaugh, or a 7'1" NBA player, the active range is close to 1/2 meter.

h_{x}
= h_{e}·( g_{e}/g_{x} )
+ L·( g_{e}/g_{x} - 1 )

= h_{e}·( g_{e}/g_{x} )
+ (½ m) ·( g_{e}/g_{x} - 1 )

= (h_{e}+½ m) ( g_{e}/g_{x} )
- ½ m

We can now compute how high Sensabaugh could jump on various objects:

GERALD SENSABAUGH JUMPING object gravity max ratio ------ --------- ------ ----- Sun 274.0m/s² -0.44m n/a Jupiter 24.79m/s² 0.16m 0.14x Earth 9.807m/s² 1.16m 1.00x Mars 3.711m/s² 3.89m 3.35x Moon 1.620m/s² 9.55m 8.23x Europa 1.315m/s² 11.9m 10.2x Ceres 0.270m/s² 60.0m 51.5x |

As we can see, even Gerald cannot jump with a gravity level as high as the sun, and he can only just barely get off the ground on Jupiter. (Technically... A platform floating over the clouds of Jupiter)

Most of us are nowhere near as fit as Gerald Sensabaugh. Let's try this again with some more down to Earth numbers. We'll assume 1/3 of a meter (13 inches) is a reasonable jump, and that force can be applied over a 1/2 meter (20 inch) range.

AVERAGE ADULT JUMPING object gravity max ratio ------ --------- ------ ----- Sun 274.0m/s² -0.47m n/a Jupiter 24.79m/s² -0.17m n/a Earth 9.807m/s² 0.33m 1.00x Mars 3.711m/s² 1.70m 5.11x Moon 1.620m/s² 4.54m 13.6x Europa 1.315m/s² 5.71m 17.1x Ceres 0.270m/s² 29.7m 89.3x |

As we can see, even Jupiter is too much gravity for most of us to handle. Even if we could stand, jumping is impossible.

Let's get back to the original question:

A world class athlete on Earth could easily jump 3.6 times as high on Mars, 8 times as high on the Moon, and nearly 50 times as high on Ceres.

A healthy adult on Earth could easily jump 5 times as high on Mars, 13 times as high on the Moon, and nearly 90 times as high on Ceres.

So for a typical adult, the results are nearly double what one might expect.