The Physics of Superheroes: Spectacular Second Edition by James Kakalios Page A
Book: The Physics of Superheroes: Spectacular Second Edition by James Kakalios
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Authors: James Kakalios
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bigger the change, the bigger the force. If Superman has a mass of 100 kilograms, then the force needed to enable him to vertically leap 660 feet is F = ma = (100 kilograms) × (250 meters/sec 2 ) = 25,000 kilograms meters/sec 2 , or about 5,600 pounds.
Fig. 5. Panels from Action Comics # 23, describing in some detail the process by which Superman is able to achieve the high initial velocities necessary for his mighty leaps.
Is it reasonable that Superman’s leg muscles could provide a force of 5,600 pounds? Why not, if Krypton’s gravity is stronger than Earth’s, and his leg muscles are able to support his weight on Krypton? Suppose that this force of 5,600 pounds is 70 percent larger than the force his legs supply while simply standing still, supporting his weight on Krypton. In this case, Superman on his home planet would weigh 3,300 pounds. His weight on Krypton is determined by his mass and the acceleration due to gravity on Krypton. We assumed that Superman’s mass is 100 kilograms, and this is his mass regardless of which planet he happens to stand on. If Superman weighs 220 pounds on Earth and nearly 3,300 pounds on Krypton, then the acceleration due to gravity on Krypton must have been fifteen times larger than that on Earth.
So, just by knowing that F = ma , making use of the definitions “distance = speed × time” and “acceleration is the change in speed over time,” and the experimental observation that Superman can “leap a tall building in a single bound,” we have figured out that the gravity on Krypton must have been fifteen times greater than on Earth.
Congratulations. You’ve just done a physics calculation!
2
DECONSTRUCTING KRYPTON— NEWTON’S LAW OF GRAVITY
NOW THAT WE HAVE DETERMINED THAT in order for Superman to leap a tall building, he must have come from a planet with a gravitational attraction fifteen times that of Earth, we next ask: How would we go about building such a planet? To answer this, we must understand the nature of a planet’s gravitational pull, and here again we rely on Newton’s genius. What follows involves more math, but bear with me for a moment. There’s a beautiful payoff that explains the connection between Newton’s apple and gravity.
As if describing the laws of motion previously discussed and inventing calculus weren’t enough, Isaac Newton also elucidated the nature of the force that two objects exert on each other owing to their gravitational attraction. In order to account for the orbits of the planets, Newton concluded that the force due to gravity between two masses (let’s call them Mass 1 and Mass 2 ) separated by a distance d is given by:
where G is the universal gravitational constant. This expression describes the gravitational attraction between any two masses, whether between the Earth and the sun, the earth and the moon or between the Earth and Superman. If one mass is the Earth and the other mass is Superman, then the distance between them is the radius of the Earth (the distance from the center of the Earth to the surface, upon which the Man of Steel is standing). For a spherically symmetric distribution of mass, such as a planet, the attractive force behaves as if all of the planet’s mass is concentrated at a single point at the planet’s core. This is why we can use the radius of the Earth as the distance in Newton’s equation separating the two masses (Earth and Superman). The force is just the gravitational pull that Superman (as well as every other person) feels. Using the mass of Superman (100 kilograms), the mass of the Earth, and the distance between Superman and the center of the Earth (the radius of the Earth), along with the measured value of the gravitational constant in the previous equation, gives the force F between Superman and the Earth to be F = 220 pounds.
But this is just Superman’s weight on Earth, which is measured when he steps on a bathroom scale on Earth. The cool thing is that these two expressions
Fig. 5. Panels from Action Comics # 23, describing in some detail the process by which Superman is able to achieve the high initial velocities necessary for his mighty leaps.
Is it reasonable that Superman’s leg muscles could provide a force of 5,600 pounds? Why not, if Krypton’s gravity is stronger than Earth’s, and his leg muscles are able to support his weight on Krypton? Suppose that this force of 5,600 pounds is 70 percent larger than the force his legs supply while simply standing still, supporting his weight on Krypton. In this case, Superman on his home planet would weigh 3,300 pounds. His weight on Krypton is determined by his mass and the acceleration due to gravity on Krypton. We assumed that Superman’s mass is 100 kilograms, and this is his mass regardless of which planet he happens to stand on. If Superman weighs 220 pounds on Earth and nearly 3,300 pounds on Krypton, then the acceleration due to gravity on Krypton must have been fifteen times larger than that on Earth.
So, just by knowing that F = ma , making use of the definitions “distance = speed × time” and “acceleration is the change in speed over time,” and the experimental observation that Superman can “leap a tall building in a single bound,” we have figured out that the gravity on Krypton must have been fifteen times greater than on Earth.
Congratulations. You’ve just done a physics calculation!
2
DECONSTRUCTING KRYPTON— NEWTON’S LAW OF GRAVITY
NOW THAT WE HAVE DETERMINED THAT in order for Superman to leap a tall building, he must have come from a planet with a gravitational attraction fifteen times that of Earth, we next ask: How would we go about building such a planet? To answer this, we must understand the nature of a planet’s gravitational pull, and here again we rely on Newton’s genius. What follows involves more math, but bear with me for a moment. There’s a beautiful payoff that explains the connection between Newton’s apple and gravity.
As if describing the laws of motion previously discussed and inventing calculus weren’t enough, Isaac Newton also elucidated the nature of the force that two objects exert on each other owing to their gravitational attraction. In order to account for the orbits of the planets, Newton concluded that the force due to gravity between two masses (let’s call them Mass 1 and Mass 2 ) separated by a distance d is given by:
where G is the universal gravitational constant. This expression describes the gravitational attraction between any two masses, whether between the Earth and the sun, the earth and the moon or between the Earth and Superman. If one mass is the Earth and the other mass is Superman, then the distance between them is the radius of the Earth (the distance from the center of the Earth to the surface, upon which the Man of Steel is standing). For a spherically symmetric distribution of mass, such as a planet, the attractive force behaves as if all of the planet’s mass is concentrated at a single point at the planet’s core. This is why we can use the radius of the Earth as the distance in Newton’s equation separating the two masses (Earth and Superman). The force is just the gravitational pull that Superman (as well as every other person) feels. Using the mass of Superman (100 kilograms), the mass of the Earth, and the distance between Superman and the center of the Earth (the radius of the Earth), along with the measured value of the gravitational constant in the previous equation, gives the force F between Superman and the Earth to be F = 220 pounds.
But this is just Superman’s weight on Earth, which is measured when he steps on a bathroom scale on Earth. The cool thing is that these two expressions
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