A team of astronauts is on a mission to land on and explore a large asteroid. In

Question

A team of astronauts is on a mission to land on and explore a large asteroid. In addition to collecting samples and performing experiments, one of their tasks is to demonstrate the concept of the escape speed by throwing rocks straight up at various initial speeds. With what minimum initial speed vesc will the rocks need to be thrown in order for them never to "fall" back to the asteroid? Assume that the asteroid is approximately spherical, with an average density ? = 4.10 × 106 g/m3 and volume V = 2.86 × 1012 m3. Recall that the universal gravitational constant is G = 6.67 × 10-11 N·m2/kg2.

A team of astronauts is on a mission to land on and explore a large asteroid. In addition to collecting samples and performing experiments, one of their tasks is to demonstrate the concept of the escape speed by throwing rocks straight up at various initial speeds. With what minimum initial speed Vesc will the rocks need to be thrown in order for them never to "fall" back to the asteroid? Assume that the asteroid is approximately spherical, with an average density = 4.10 x 106 g/m2 and volume V: 2.86 x 102 m3 Recall that the universal gravitational constant is G 6.67 x10-1N m2lkg2 Number m/s esc

Explanation / Answer

This is a problem where we have to determine the escape velocity of the asteroid. This is the velocity at which the astronauts would have to throw the stones.

Escape velocity in m/sec can be calculated using

Step 1: We need to determine the mass M of the asteroid.

In order to find mass when you have density and volume, you use the relationship

Density = Mass / Volume

Or Mass = Density x Volume

M = 4.10 × 106 g/m3    x     2.86 × 1012 m3

M = 11.73 x 1018 g

Convert the M to kg by multiplying and dividing by 103, we get M = 11.73 x 1015 Kg

Step 2: We need to determine the radius r of the ‘approximately spherical’ asteroid.

Given the volume of the sphere, we can use,

So solving for r, we have

Plugging in the values of V = 2.86 × 1012 m3, we get r = 8807.1 m

Step 3: With value of M and r known, we can now determine the Vescape using

M = 11.73 x 1015 Kg

G = 6.67 × 10-11 Nm2/kg2

r = 8807.1 m

Vescape = sqrt (177.67) = 13.33 m/s

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