The asteroid Eros, one of the many minor planets that orbit the Sun in the region between Mars and Jupiter, has a radius of \(7.0 \mathrm{~km}\) and a mass of \(5.0 \times 10^{15} \mathrm{~kg} .(a)\) If you were standing on Eros, could you lift a \(2000-\mathrm{kg}\) pickup truck? \((b)\) Could you run fast enough to put yourself into orbit? Ignore effects due to the rotation of the asteroid. (Note: The Olympic records for the \(400-\mathrm{m}\) run correspond to speeds of \(9.1 \mathrm{~m} / \mathrm{s}\) for men and \(8.2 \mathrm{~m} / \mathrm{s}\) for women.)

Short Answer

Expert verified
For part (a), assuming you can lift a certain weight on Earth, you would be able to lift a heavier weight on Eros due to the lower gravitational force. For part (b), the escape velocity on Eros is much higher than a human's typical running speed, so you cannot run fast enough to put yourself into orbit.

Step by step solution

01

Calculate the gravitational force exerted by Eros

We can use the equation for gravitational force \( F_g = G * m1 * m2/r^2 \) where \( G = 6.674 * 10^{-11} N(m/kg)^2 \), m1 is the mass of the pickup truck (2000 kg), m2 is the mass of Eros \( 5.0 * 10^{15} kg \), and r is the radius of Eros (7000 m). After calculating, the gravitational force will be smaller than on earth.
02

Determine if you can lift a pickup truck on Eros

On Earth, the force experienced by a 2000-kg object is \( F = m * g \) where g is acceleration due to gravity (9.8 m/s^2). If the force needed to lift the truck on Eros is less than the force experienced by the truck on Earth, then it could be lifted on Eros. So, find that force and make a comparison.
03

Calculate the escape velocity on Eros

The equation for escape velocity is \( v = \sqrt{2*G*M/r} \), where G is the gravitational constant, M is the mass of the celestial body, and r is the distance from the center of the celestial body (or radius for planets and asteroids). Calculate the escape velocity and compare it to a human's running speed.
04

Determine if you can put yourself into orbit

If the calculated escape velocity on Eros is lower than the average speed of an Olympic runner (9.1 m/s for men and 8.2 m/s for women), then it would be possible for a human to run fast enough to reach escape velocity and put themselves into orbit.

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