Finding Escape Speed on Mars: Advanced Gravitational Potential Energy

[mutineer123]

Homework Statement

The mean diameters of Mars and Earth are 6.9 10^3 km and 1.3 10^4 km, respectively.?
The mass of Mars is 0.11 times Earth's mass.
(a) What is the ratio of the mean density of Mars to that of Earth?

(b) What is the value of the gravitational acceleration on Mars?

(c) What is the escape speed on Mars?

Homework Equations

The Attempt at a Solution

I got 1 and 2, but am stuck in 3. ANSWER 1 is 0.74, ANSWER 2 is 3.8m/s^2.

For The escape speed on Mars, I am not sure if I entirely understand this question even! One of my friends told me that escape speed is when
1/2 m v^2 > mgh. And using this method I am getting the right answer, but his definition of escape speed is not entirely satisfying. You know the formula a=v^2/r right? Why can't we use that to get V here?? Also, is there a way of finding out the answer using formulas like g=Gm/r^2 ?

 

[azizlwl]

For excape, r=∞

$\int_R^∞ \! f(r) \, \mathrm{d} r$

$\int_R^∞ \! \frac {GMm}{r^2} \, \mathrm{d} r$

ΔPE=GMm/R=(GMm/R²).R=mgR

 

[mutineer123]

For excape, r=∞

$\int_R^∞ \! f(r) \, \mathrm{d} r$

$\int_R^∞ \! \frac {GMm}{r^2} \, \mathrm{d} r$

ΔPE=GMm/R=(GMm/R²).R=mgR

Edit:
By the definition of U:-W=ΔU

I have no idea what you did after your integration. Can u explain?

 

[azizlwl]

It is the total energy needed to take an object from the radius R to infinity.
Since the final velocity is zero, energy conservation gives
ΔKE+ΔPE=0

ΔPE=(GMm/R).(R/R)

 

[mutineer123]

It is the total energy needed to take an object from the radius R to infinity.
Since the final velocity is zero, energy conservation gives
ΔKE+ΔPE=0

ΔPE=(GMm/R).(R/R)

How is the final velocity 0?
ANd uve written at the beginning "For excape, r=∞', why do we take that assumption?

 

[mutineer123]

Are you sure your method will works? u say ΔKE+ΔPE=0 where ΔPE=(GMm/R).(R/R). But you can't use the formula ΔPE=(GMm/R).(R/R) in this way, you don't know the mass of mars!

 

[ehild]

Escape means to run away. An object -a spaceship- escapes Mars if it is not confined to orbit around it, but can go away from its gravitational field, that is, to infinity.
You know how the mass of Mars is related to the mass of Earth. You also know the radii, both of Mars and Earth. And you know the gravitational acceleration at the surface of Earth.

ehild

 

[mutineer123]

Escape means to run away. An object -a spaceship- escapes Mars if it is not confined to orbit around it, but can go away from its gravitational field, that is, to infinity.
You know how the mass of Mars is related to the mass of Earth. You also know the radii, both of Mars and Earth. And you know the gravitational acceleration at the surface of Earth.

ehild

I also know mar's gravitational acceleration. But like i said using GMm/r cannot yield anything because like you know, the masses are in ratio. So I don't know the mass! How can I proceed then??

 

[ehild]

Do you know the mass of Earth?
The gravitational acceleration on the Earth surface is g=GM/R² (M is the mass of Earth and R is the radius of Earth). g=9.81 m/s². The diameter of the Earth is given in the problem.
The mass of Mars is 0.11 M_earth.
ehild

 

[mutineer123]

Do you know the mass of Earth?
The gravitational acceleration on the Earth surface is g=GM/R² (M is the mass of Earth and R is the radius of Earth). g=9.81 m/s². The diameter of the Earth is given in the problem.
The mass of Mars is 0.11 M_earth.
ehild

Yes yes, that substitution escaped me...I did it that way now. But i am not getting the same answer as i did in the mgh way...any suggestions on why?

 

[ehild]

PE=mgh is an approximate formula valid only very near to the surface of the Earth, and it is the potential energy with respect to the ground.

GmM/R is the general gravitational potential energy of two point-like or spherical objects.

ehild