Asteroids have average densities of about 2,615 kg/m^3 and radii from 470 km dow
ID: 2017653 • Letter: A
Question
Asteroids have average densities of about 2,615 kg/m^3 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 your 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.
Answer should be: 3.66 km
Explanation / Answer
Asteroids have average densities of about 2,615 kg/m^3 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 your jump speed by relating it to the maximum height that you can jump on earth, for this problem assume that height in 1 meter).
When you're jumping on earth, suppose you start with a u velocity. If the height you jump is 1 m,
v = u + at
0 = u - 9.8t
u = 9.8t
t = u/9.8
s = ( v + u ) / 2*t
1 = ( 0 + u ) / (2*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. ( this is a fact ) As you're in the largest asteroid, at infinity your velocity is also zero, therefore 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)*4.427^2 = G*([4/3] r^3*) / r
(1/2)*4.427^2 = G*(4 r^2*2615) / 3
9.7991645 = 17.0457 * G r^2
9.97991645 = 17.0457*6.67*10^(-11) r^2
r = 3.66 km
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