In the 2004 Olympic Games, the men\'s high-jump medalists cleared a height of at

Question

In the 2004 Olympic Games, the men's high-jump medalists cleared a height of at least 2.34m. There are numerous small asteroids (radii of just a few kilometers' assume them to be spherical) orbiting the Sun between Jupiter and Mars. could one of these athletes actually jump off an asteroid, never to return? What size is the largest asteroid from which an Olympian could jump off? Based on your calculations, a new event may be initiated for the 2040 Summer Games to be hosted by the Mars Colony. You may make these assumptions: The high jumper can jump at the same initial velocity on the asteroid as on Earth, and the asteroid has the same average density as the Earth.

(Essential University Physics Richard Wolfson Volume 1 Ch 8 #65)

Explanation / Answer

The answer is Yes. The high - jump medalists can easily jump off an asteroid. Now let us find out the size (volume) of the asteroid. He jumps the height, h = 2.34 m But we have the formula for the maximum height by an upword projected body is, u^2/2g = h From the above, u = sqrt (2gh) = sqrt (2 * 9.8 * 2.34) = 6.8 m/s As we discuss above the Olympian could jump off the asteroid means this initial velocity is the escape velocity of the person on that asteroid. So the escape velocity on the asteroid, v = 6.8 m/s But we have a formula for the escape velocity, v = sqrt (2gR) Where the R = The radius of the asteroid Then, sqrt (2gR ) = 6.8 m/s                  2gR = 45.86                     R = 45.86/2g = 2.34 m So the volume of the spherical asteroid, V = (4/3)r^3 = 53.64 m^3 So the size (volume) of the asteroid, v = 53.64 m^3

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