physics paper

PHY110 Understanding the Natural World

Fall 2014

Example term paper.

How high can you jump on Mars?

Introduction

Since the first man-made satellite Sputnik 1 was successfully launched in 1957 [1 ] space exploration has been a landmark of human technological achievement and a source of national pride for a number of countries including the United States, the former U.S.S.R. and China. Next to Earth, Mars is the most hospitable planet for life in the solar system – it has polar ice caps, seasons and there is geological evidence of the existence of liquid water long ago [2]. Exploration of Mars not only satisfies human curiosity about extra-terrestrial life, it also answers important questions about the history of the Earth, its climate and the origins of life. NASA has invested significant resources in more than sixteen Mars-related missions in the past forty years, including orbiting spacecraft missions (Viking 1 and 2, Mars Observer, Mars Climate Orbiter, Mars Odyssey) as well as landing rover missions. (Mars Pathfinder, Mars Polar Lander, Spirit, Opportunity and Curiosity) [1]. Just days ago, NASA Jet Propulsion Laboratory’s latest rover mission, Curiosity, has arrived at its destination, a large mountain called Mount Sharp, after two years of traveling across the Martian surface. The current Mars Exploration Program is focused on “Seeking Signs of Life”, namely looking for microbes and evaluating the habitability of the Red Planet. The understanding of physical conditions on other planets offers possibilities for human colonization in the far future. If humans were to inhabit Mars, one important question would be how our locomotion be affected by the changed gravity, thin atmosphere and geological landscape. To take a basic step toward understanding this problem, we will examine the special case of a human jumping vertically on Mars.

Assumptions and Calculations

Mars is the fourth planet from the Sun, with a mean radius of 3,389.5 km and mass of about one-tenth that of the Earth [3]. Figure 1 [2] shows the relative size scale of Earth and Mars side by side. The length of a day on Mars is about 24.6 hours[3] and the length of a year (orbit around the sun) is about 1.88 Earth years [3]. Thus, there are some characteristics of Mars which are similar to Earth. However, the extremely thin atmosphere, lack of an ozone layer, and extreme temperatures make Mars dramatically different from Earth. Human locomotion is expected to depend highly on factors such as the pull of gravity on the surface of Mars, atmospheric pressure, and the properties of the geology on its surface.

Figure 1 [2] Images of the Earth and Mars placed side by side to illustrate relative size

To calculate how high a human being a jump on Mars, we will use the simple principle of conservation of mechanical energy and several assumptions about the surface of Mars. Mars has a very thin atmosphere composed of Carbon Dioxide, Nitrogen and Argon. Since we neglect air friction when jumping on the surface of the Earth, we will also discount air friction for the surface of Mars. NASA reports the gravitational constant on the surface of Mars to be 3.71 m/s2 [3], in contrast to Earth’s 9.8 m/s2. For simplicity, we let the average mass of a person to be about 70 kg (154 lbs). To estimate the height that a person can jump on Mars, we will use the principle of energy conservation. Hobson chapter 6 [4] tells us that energy cannot be created or destroyed, only transformed from one form to another. At the instant a person leaps off the ground, he or she has not gained any height and all of the energy is in the form of kinetic energy (KE). This is transformed to gravitational potential energy as the person gains height. At the top of the jump, all of the person’s energy is in the form of gravitational potential energy (GPE). This is shown in Figure 2 a and b.

Figure 2a. Immediately after leaping off the ground, a person’s energy is all kinetic energy (KE). Figure 2b. At the top of the jump, a person’s energy is all gravitational potential energy (GPE).

The speed at which a human can jump off the ground is determined by the energy in of his or her muscle tissues, anatomy and conditioning. A 2001 study published in the American Journal of Physics tested a skilled jumper’s movements using a force platform and found his maximum velocity to be about 3 m/s [5]. Using the assumptions above, a typical person’s kinetic energy is estimated to be

KE=½mv2=½(70 kg)(3 m/s)2=315 J

Since energy is conserved and air friction is neglected, all of the person’s kinetic energy is transferred to gravitational potential energy at the top of their jump. We can use this to solve for h, the maximum height reached. However, instead of Earth’s acceleration due to gravity g, we must use the gravitational acceleration of Mars, 3.71 m/s2, which we will call gMars.

mgMarsh=315 J

((70 kg) (3.71 m/s2)h=315 J

( h=1.2 meters

The above equation yields a value of around 1.2 meters, so a typical person on Mars can do a vertical jump of almost 4 feet! This is much higher than the average NBA player’s vertical leap of 0.7 meters [6]. However, the result is perhaps not surprising, since our physical intuition says a person should be able to jump higher due to the smaller gravitational pull on Mars.

Discussion and Conclusions

Here, we have used a simple mechanical energy conservation model to estimate how high a 70 kg human being can jump on Mars. By equating kinetic to potential energy and using the reduced acceleration due to gravity on the Martian surface, the results show that a typical human could jump over a meter. This simple estimate of how high a person can jump on Mars does not take into account several important factors. First of all, the height of a person’s jump should depend on his or her footwear. A person in bare feet would have a definite disadvantage in jumping compared to a person in sneakers. What kinds of shoes are suitable for space exploration is a subject that merits its own discussion. Secondly, how fast a person propels off the ground will depend on the terrain. Mars is a rocky, or terrestrial planet, with a solid surface of volcanoes and craters. In this sense, jumping off a rocky landscape would be similar to jumping off the surface of the Earth.

Despite the fact that vertical jumping is theoretically possible and horizontal locomotion has already been demonstrated by Mars rovers, whether or not Mars has ever been, or will ever be habitable, remains an open question. The current geologic record has evidence related to the existence of liquid water, but the search for organics, the molecules that make up living matter, has not yet yielded definite findings. There is scientific evidence that Mars may have been much warmer and wetter than it is now, but its current oxygen-lacking atmosphere, extreme temperatures, and lack of protection from space radiation make it very hostile to humans. If we choose to travel to Mars and live there, we could become “life on Mars”. Living on Mars would most certainly mean living indoors, relying on technology to artificially provide oxygen, heat, pressure, and protection from damaging radiation. Even with immediate survival ensured, it’s unclear if humans could live on Mars. In the presence of a much weaker gravitational field, how would the human body react and grow? Where would we get our food sources? Shipping items from Earth could take the better part of a year. What would be the psychological consequences of living on such a barren environment? It would not be easy, but some, like Canadian inventor Elon Musk, have already started to envision it [7]. Such space pioneers are motivated by a desire to achieve what is hard and the drive to explore. It’s even possible that if Earth becomes inhospitable in the future, we would need another planet to live on. Until then, Mars exploration will remain an exciting and challenging scientific frontier.

[1] http://www.rocketmime.com/space/timeline.html

[2] http://mars.nasa.gov/programmissions/science/

[3] http://solarsystem.nasa.gov/planets/profile.cfm?Display=Facts&Object=Mars

[4] A. Hobson, Physics: Concepts and Connections. Boston: Addison-Wesley (2010)

[5] N. Linthorne, “Analysis of standing vertical jumps using a force platform,” Am. J. Phys. 69 pp. 1198-1204 (2001).

[6] http://www.topendsports.com/testing/results/vertical-jump.htm

[7] http://www.universetoday.com/111462/how-can-we-live-on-mars/

b�

a

All GPE

h=0, all KE

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