Estimate the radius of the largest asteroid you could escape by jumping

[leonne]

Homework Statement

estimate the radius of the largest asteroid from which you could escape simply by jumping off.

Homework Equations

R^2= 3gh/4piGp

The Attempt at a Solution

Would I use this formula to solve it? Found it online but dose not seem right. this is a 2 part problem first one asked for the velocity you get when you jump up and got 1.72e6 cm/s

 

[JaredJames]

This question is dependent on the mass of the asteroid (giving the gravitational force).

The lower the mass, the smaller the radius you could jump from to escape its gravitational pull.

Without knowing the mass of the asteroid, you cannot give a radius.

EDIT: I would also avoid using cm/s. You should work in SI units - m/s. Note, the velocity during a jump would not be constant (m/s) you would slow on the initial ascent, stop, and then descend so you should have a value for the initial acceleration of the jump in (m/s^2). On earth, your initial jump acceleration must be > 9.81m/s^2 in order to get you off the floor.

The relevant equation you provided uses g (gravitational acceleration). Although I would ignore this equation and go with the one below.

g = ( G x M1 x M2 ) / r^2

Where: G is the gravitational constant, M1 is the mass of the asteroid, M2 is the mass of the person jumping and r is the radius of the asteroid.

Once you know all those factors you can calculate the acceleration due to gravity acting on the person. Once you have that, you know what acceleration you need to escape the asteroids gravity.

 

[leonne]

ok thxs ill try it out

 

[leonne]

Hey just wonder could you use the escape velocity to find out? v=(2GM/r)^1/2 v would be the velocity i figured out jumping on Earth then plug in the other and solve for r

 

[JaredJames]

Again, you have M there. This is the mass of the asteroid. You need a mass for the asteroid, once you have this, we can plug in G, M and v and rearrange to get r.

Is that everything you were given in the question?

 

[leonne]

well it says you will need to make an assumption about the mean density that's why I used that first equation cause it had density, but i guess does not matter much

 

[JaredJames]

Ah so there was more to it. If you have a radius and know the mean density for an asteroid, you can calculate the mass.

 

[JaredJames]

I would use the equation v_escape = Square Root ( ( 2GM ) / r )

Where v_escape = your jump speed, G = gravitational constant, M = mass of asteroid, r = radius

M = volume x density

v_escape = Square Root ( ( 2GVD ) / r )

Where V = volume, D = mean density

Now assume the asteroid is a sphere to simplify things, substitute in the formula for the volume of a sphere instead of V in the above. From there you should be able to see how you can rearrange to get the answer. Let me know how you get on.

 

[leonne]

k thxs got it

 

[JaredJames]

What value did you get for r?

 

[JaredJames]

I have to question, do you believe you can jump at 17200m/s on Earth? How did you calculate this value.

 

[leonne]

o um i used the conservation on energy pei +kei = pef kef found that pe= mgh then something about transferring the same energy into kinetics. didnt take general physics since 3 years so forgot lol then did 0 +kei = 3e8 +3e8 solve for kei then plug that in kinetics to find v also for avg mass i found that all the asteroid in the solar system have a mass about .o2 of the moon so found the mass of all then found there is about 1mill asteroid so divided that to find mass of 1

edit ok i think i see the problem when i found kei i got 9e16 but did something wrong and should be 6e8

 

[JaredJames]

You need to use the average density of an asteroid, not the average mass. Your question does say to use mean density of an asteroid.

Once you have the average density, you can plug it into the equation I gave above (and following my outline) you can calculate the radius.

 

[JaredJames]

To calculate your initial jump velocity, you either need the height you jumped or the time you were in the air.

If you know t:
Using v = u + at, rearrange to give u = ?

If you know height s:
Using v^2 = u^2 + 2as, rearrange to give u = ?

Where v = final velocity, u = initial velocity, a = acceleration, t = time in air

Remember, acceleration will be opposite to the direction of travel and t = the time taken from start to peak jump height, not the entire jump up, stop down.

Once you've done that, you have your v for my previous equation.

Plug all the values in and rearrange to give you the r value based on your maximum possible escape velocity.

 

[leonne]

ok thanks