Will colonists on Mars have enhanced physical mobility relative to an Earth-normal environment?

Question

A Princess of Mars (1912) by Edgar Rice Burroughs in Chapter III describes this "superhuman leap" by the hero John Carter:

My effort was crowned with a success which appalled me no less than it seemed to surprise the Martian warriors, for it carried me fully thirty feet into the air and landed me a hundred feet from my pursuers and on the opposite side of the enclosure.

The novel was written at a time when the possibility of breathable air on Mars still existed; knowledge of the lesser gravity of Mars spawned such assumptions.

Will Humans from Earth really possess the ability to leap 30 feet high and land 100 feet away on Mars?

Answer 1

Let's first see what calculations we have available on the Internet, maybe from what could be considered relatively reliable sources:

University of Arizona - Phoenix Mars Mission - Mars 101:
Gravity on Mars is only about 38% of Earth's. So, if you weighed 100 pounds on Earth, you would only weigh about 38 pounds on Mars. And if you can jump one meter (3.3 feet) high on Earth, you would be able to jump 2.64 meters (almost 9 feet) high on Mars. The lower gravity on Mars could prove beneficial to future astronauts, as it would permit them to easily walk around the surface wearing large space suits and carrying heavy backpacks.

But let's put this claim to the test. Not because we don't trust their calculations, but because we want to understand the physics involved a bit better than that description provides. We could assume the following;

If you can jump 1 m high in Earth's gravity (distance between feet and the ground when feet are stretched in the air), your initial velocity (or linear momentum) would have to be 4.43 m/s. This is the constant that wouldn't change no matter at what gravitational conditions you jump, as it is the ability of your body to produce momentary force, in our case in the vertical direction required for a jump, producing momentum force in the opposite direction to gravitational acceleration.

Saying it differently, your legs can't spring any faster just because the gravitational acceleration working on them is smaller. They can't, because of your muscles and bones only being capable of producing that much force in that much time. Point in case, if this wasn't so, astronauts on the IIS would be able to move their limbs at the speeds that even slow-motion cameras would have hard time recording. And it isn't the atmospheric pressure (drag, or air resistance) slowing you down much either, not in our case at least (and even less so on Mars, with atmospheric density equal to the density found 35 km above the Earth's surface), otherwise those ISS astronauts would be kicking really lively during extravehicular activity (EVA).

OK, I hope I demonstrated well enough, that we can consider this 4.43 m/s as constant. So let's take this number and gravitational acceleration on Mars (0.38 of Earth's) into our own calculations, and we get 2.633135 m for the height of our jump, and 2.3776 seconds "in the air". Accounting for a small margin of error and rounding up of the end results, the calculations on the page of the University of Arizona are indeed correct.

But how far could you jump? Ballistics teach us that the best angle to achieve the longest range is 45°, if we don't have to consider aerodynamics. Since we don't have to consider aerodynamics at all (our bodies are hardly airworthy at such low speeds and the atmosphere on Mars is utterly neglectable in our case anyway), this is really convenient, because we can apply simple trigonometry to our previous results, where the hypotenuse length in a real triangle is those 2.633135 m from before, when we jumped straight up. And we know that hypotenuse c² = sqrt(a² + b²), where a and b legs are of equal length in a right triangle with the hypotenuse at a 45° angle. A bit of basic arithmetic, and we end up with the calculated length of our "ballistically best" hop range of 3.723815 m (plus as far as you can stretch your legs and pull yourself back up on landing without falling on your back, what basically athletes do with their long jumps). This is of course a jump from a stationary position, or a hop, and not with a run before the jump, gaining horizontal momentum.

Where the quoted author gets those numbers from, is however beyond my knowledge. Maths aren't all that sound anyway; If we apply our hop logic from before (l = 2(sqrt(c²/2))) to those "30 ft high", and don't assume humans living on Mars in time grow wings, we get a distance of 42.43 feet (plus slightly more with stretching legs forward and pushing oneself back in the upright position upon landing). If the author didn't mean "hops" but actual "long jumps" with a run before the jump, then it becomes a problem of jump technique, which I'm afraid I can't calculate for conditions on Mars, but I presume the major problem in gaining much speed than on Earth before the jump on Mars would be one of traction due to lesser gravity, and I highly doubt jumps much longer than 30 ft in length would be possible for an average person (the one we assumed a high jump of feet 1 m off the ground when stretched in our previous calculations), and maybe roughly double as much (~ 60 ft) for a top athlete.

For comparison, current long jump world record holder Mike Powell (USA) managed 8.95 m (29 ft 4¼ in) in Tokyo, back in 1991. If we multiply this value with 1/0.38 (or 2.6316 to approximate conditions on Mars based on 38% of Earth's gravity), we get 23.55 m (or 77 ft 3¹⁷⁄₆₄ in), which sounds about right, if we take some non-negligible percentage off it due to loss in traction. I.e., we get back to roughly 60 ft (let's sum it up to 20 meters) for a top athlete.

Then again, that work you're quoting is fiction, and I'm making too much of it again... ;)

Answer 2

The initial velocity assumption in TidalWave's answer is incorrect.

The maximum initial velocity value is calculated from the accelerating force (muscle work) minus the decelerating force (gravity) over the time of the acceleration (time before leaving the ground). If the decelerating force (gravity) is smaller, the vector force is higher, the time before leaving the ground shorter and the initial velocity higher.

So it's not a constant. In lower gravity initial velocity is higher. Same goes the other way around, in higher gravity initial velocity would be lower until the point where it's so low you can't leave the ground at all.

To make this more obvious try a straight kick forward while sitting down. The acceleration of your leg in such a movement is significantly faster than when jumping off the ground since no gravitational force pulling on your body weight is holding you back from maximizing the speed of which you extend your leg.

There is of course a mechanical limitation to how fast you can contract your muscles but boxers for instance have achieved hand speeds at full extension of over 10 m/s so human muscles can contract at speeds in extremities to achieve maximum speeds way above 4.43 m/s. The truth would be an initial speed somewhere below 10 m/s but above 4.43.

If you build a horizontal machine where you can push of a wall and measure the initial speed, you can compare this to your jumping initial speed and get some kind of estimate.