Calculating Max Radius of Asteroid to Escape From

[Master J]

I need to estimate the maximum radius of an asteroid that I could jump free from.

All I am told is that its density is the same as that of the earth, and I am given a figure for the Earth's radius.

I know the equations for the escape velocity etc, but I just can't seem to get around this one. I keep ending up with 2 unknowns no matter which way I try it.

Any hints??

 

[SammyS]

Be more specific with what you have tried & where you are stuck.

 

[Master J]

Escape velocity is

v = SQRT 2gr

since the densities are the same, this gives me the ratio of escape velocities of the asteroid and Earth is equal to the ratio of their radii. I can work out the escape velocity of the earth, O have its radius, yet I am left with the escape velocity of the asteroid AND its radius, 2 unknowns??

 

[gomunkul51]

Basically you need to estimate your initial velocity when you jump i.e. the velocity after you extended you legs and developed momentum.

You can do this by estimating where is the center of mass of your body and measuring it's height above ground, then jump as hard as you can and estimate what is the height difference of your center of mass at you max height and its original position.

The use "a Conservation Law" to find the initial velocity from the data above.

remember that the impulse is (force - mg)*(delta time) !

probably you need to do some mathematics to do the correct estimation also and you need to estimate the time it take to apply the force with you legs (that's basically the way i solved it).

another way: you can estimate what impulse you can develop in your jump i.e. what force to you apply with your legs when you are preforming a jump and for how long do you apply it.

good luck.

 

[Master J]

When i jump on the Earth or asteroid??

 

[gomunkul51]

If you have an asteroid to jump on then yes, otherwise the earth.
I advise thinking about the question before writing equations and planing the way you will going to solve it, I provided a way to attack the problem.
Think about how can you quantize your "jump".

 

[Master J]

Ok so on earth, when I jump with initial speed v, I can get v from

mgh = 1/2 mv^2 , if I take an estimate of h.

SO then, on the asteroid, I jump with this speed, but now I escape from the asteroid, so this is my escape velocity ( no smaller than).

so v = SQRT 2gr. But again, I can't find g (of the asteroid).

I seem to be goin in circles here :/

 

[SammyS]

$$g=\frac{G\,M_E}{{R_E}^2}$$

and $$M_E=\rho\ \frac{4}{3}\pi{R_E}^3$$

∴ $$g=\frac{4\,G\rho\,\pi\,R_E}{3}$$

Of course that's for Earth. A similar result holds for an asteroid with density ρ .