2AI Labs | Speedrun | HOW HIGH CAN YOU JUMP ON MARS?

HOW HIGH CAN YOU JUMP ON MARS?

And while we are at it...

Thousands of online articles address this question, and even university physicists agree it is easy to resolve. Gravity on Earth is 2.64 times stronger on Earth than on Mars, therefore you can jump 2.64 times higher on Mars.

Amazingly, it turns out this answer is wrong by a factor of 2.

Let's see why.

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: F = m (ae+ge)

When you jump on a planet with local gravity gx your leg force is:

F = m (ax+gx)

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

ax = ae + ge - gx

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

ve2 = 2 L ae

vx2 = 2 L ax
      = 2 L ( ae + ge - gx )
      = 2 L ae + 2L ( ge - gx )
      = ve2 + 2L ( ge - gx )

Suppose the maximum height an athete can reach by jumping on Earth is he. We can write an upper bound on ve in terms of he and ge:

ve2 ≤ 2 · he · ge

And therefore:

vx2 ≤ 2 · he · ge + 2 · L · ( ge - gx )

The maximum height of a jump on a planet with gravity gx is threfore:

hx ≤ [ 2 he ge + 2 L ( ge - gx ) ] / (2gx)
      = he ge / gx + L ge / gx - L gx / gx
      = he·( ge/gx ) + L·( ge/gx - 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.

hx = he·( ge/gx ) + L·( ge/gx - 1 )
      = he·( ge/gx ) + (½ m) ·( ge/gx - 1 )
      = (he+½ m) ( ge/gx ) - 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.


Dr. Timothy P. Barber / 2AI © 18 MAY 2020