Asteroids have average densities of about 3,550 kg/m3 and radii from 470 km down

Question

Asteroids have average densities of about 3,550 kg/m3 and radii from 470 km down to less than a kilometer. Assuming that the asteroid has a spherically symmetric mass distribution, estimate the radius of the largest asteroid from which you could escape simply by jumping off. (Hint: You can estimate you jump speed by relating it to the maximum height that you can jump on earth, for this problem assume that height in 1 meter).

Give your answer in kilometers to the second decimal place

The answer is 3.14

Clearly write out the steps to solve this and show the numbers plugged in at the end

Explanation / Answer

When you're jumping on earth, suppose you start with a u velocity. If the height you jump is 1m,
v = u + at
0 = u - 9.8t
u = 9.8t
t = u/9.8

s = ( v + u ) / 2 x t
1 = ( 0 + u ) / 2 x u/9.8
u = 4.427 ms^-1

On the asteroid, you jump at the same velocity.
Energy you have at the beginning = kinetic energy + gravitational potential energy
= 1/2 mv^2 + ( -GMm/r )

To escape from the asteroid's gravitation, theoreticaly you should go to infinity, and at infinity your gravitational potential energy is zero. ( that's a fact ) As you're in the largest asteroid, at infinity your velocity is also zero, therefor kinetic energy is zero. ( kinetic energy is consumed to increase the potential energy )

1/2 mv^2 + ( -GMm/r ) = 0
1/2 mv^2 = GMm/r
1/2 x 4.427^2 = G x (4/3 pi r^3 x ) / r

1/2 x 4.427^2 = G * 4 pi r^2 x 3550 / 3
9.7991645 = 19.861 * G r^2

9.97991645 = 19.861*6.67*10^(-11) r^2
r = 3.14 km

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