Gravitational Potential Energy problem-Physics1-help

[moephysics]

Homework Statement

An asteriod hits the Moon and ejects a large rock from its surface. The rock has enough speed to travel to a point between the Earth and the moon where the gravitational forces on it from the Earth and the moon are equal and opposite in direction. At that point the rock has a very small velocity toward Eart. What is the speed of the rock when it encounters Earth's atmosphere at an altitude of 700 Km above the surface?

Homework Equations

Gravitational potential energy: PE= -Gm1m2/r
Conservation of Mechanical Energy: KE(intial)+PE(intial)=KE(final)+PE(final)

The Attempt at a Solution

well firstly ididnt get what they mean that its moving towrds earth, I mean if the forces between the moon and Earth are equal and opposite in direction shouldn't the velocity of the rock be zero, Ijust need calrification here becuase that was confusing me a bit. And also I tired finding the velocity that it moves with toward Earth but I don't know anything about the distance that it travelled. I just basically need someone to clarify this problem for me that's all and thankyou:).

 

[Hootenanny]

Homework Statement

An asteriod hits the Moon and ejects a large rock from its surface. The rock has enough speed to travel to a point between the Earth and the moon where the gravitational forces on it from the Earth and the moon are equal and opposite in direction. At that point the rock has a very small velocity toward Eart. What is the speed of the rock when it encounters Earth's atmosphere at an altitude of 700 Km above the surface?

Homework Equations

Gravitational potential energy: PE= -Gm1m2/r
Conservation of Mechanical Energy: KE(intial)+PE(intial)=KE(final)+PE(final)

The Attempt at a Solution

well firstly ididnt get what they mean that its moving towrds earth, I mean if the forces between the moon and Earth are equal and opposite in direction shouldn't the velocity of the rock be zero, Ijust need calrification here becuase that was confusing me a bit. And also I tired finding the velocity that it moves with toward Earth but I don't know anything about the distance that it travelled. I just basically need someone to clarify this problem for me that's all and thankyou:).

Okay, I'll try and clarify the problem for you. Imagine that the rock is traveling form the moon towards the earth. Now, when it reaches the point where there is no net force acting on the rock it still moving towards the Earth some non-zero velocity. Recall Newton's First Law, if there are no external forces acting on a body, then the body remains at rest or _________

 

[moephysics]

ohh yea! I see and one more thing when they say enough speed to travel between a point between the Earth and the moon where the gravitational forces are equal and opposite in direction, could they possibly be hinting that the rock would travel initially at the escape speed from the moon or I am I analyzing it the wrong way. that's all and thank you once more:)

 

[Hootenanny]

ohh yea! I see and one more thing when they say enough speed to travel between a point between the Earth and the moon where the gravitational forces are equal and opposite in direction, could they possibly be hinting that the rock would travel initially at the escape speed from the moon or I am I analyzing it the wrong way. that's all and thank you once more:)

I wouldn't say that the rock was launched at the escape velocity. Instead, I would say that at the point between the Earth and the moon where the gravitational forces are equal and opposite in direction, that the rock has a very small velocity - so small in fact that you could consider it to be zero.

 

[moephysics]

I wouldn't say that the rock was launched at the escape velocity. Instead, I would say that at the point between the Earth and the moon where the gravitational forces are equal and opposite in direction, that the rock has a very small velocity - so small in fact that you could consider it to be zero.

wait, but if its zero how does the rock keep moving towards Earth until it reaches an altitude of 700 km ?

 

[Hootenanny]

wait, but if its zero how does the rock keep moving towards Earth until it reaches an altitude of 700 km ?

I didn't say it was zero, I said you could consider it to be zero:

I wouldn't say that the rock was launched at the escape velocity. Instead, I would say that at the point between the Earth and the moon where the gravitational forces are equal and opposite in direction, that the rock has a very small velocity - so small in fact that you could consider it to be zero.

If it had a non-zero velocity at this point, then it would simply there forever, never moving. It does have some small velocity at this point to ensure that it doesn't just sit there but moves towards Earth instead. However, this velocity is so small that you can neglect it and assume that the rock starts moving from rest at this point.

Do you follow?

 

[moephysics]

oh ok yea I see where your coming from,, alright thanks a lot your help is appreciated,, and oh yea one more thing I've been trying to figure out the distance where this rock was undergoing zero net force,, and for some reason I am not getting any where I don't know why. I mean this is how Iam thinking of it I figure out the intial distance between the rock and the Earth then since I am going to assume that it starts from rest I use this equation
v= √2(-GMearth/7078+GMearth/Rre).

Where 7078 is the distance from the Earth center to the point of 700km altitude in km and Rre is the initial distance between the Earth and the rock. Doing that I could figure out the final velocity, but I tired finding the initial distance between rock and Earth but i don't seem to be getting any where.

 

[Hootenanny]

I think it's best if we start from scratch.

Using Newton's Law of gravitation, can you write down an equation the represents the net force felt by the rock (i.e. the sum of the forces exerted by the moon and earth)?

 

[moephysics]

umm alright yea sure, so it would be something like this

Fnet=GMeMr/Rre^2-GMoMr/Rro^2=0

where Me=mass of earth
Mr=mass of rock
Rre=distance between Earth and rock
Mo=Mass of moon
Rro=distance between moon and rock

 

[Hootenanny]

umm alright yea sure, so it would be something like this

Fnet=GMeMr/Rre^2-GMoMr/Rro^2=0

where Me=mass of earth
Mr=mass of rock
Rre=distance between Earth and rock
Mo=Mass of moon
Rro=distance between moon and rock

Looks good. So we want to figure out where abouts this F_net is zero. At the moment we have two reference points (the Earth and the moon) and we're measuring the distance from both points of reference r_e and r_o.

To make our life easier, let's pick just one reference point say the centre of the earth. Can you now re-write r_o in terms of r_e and the distance from the centre of the Earth to the centre of the moon d_eo?

Notice that you can also cancel off the mass of the rock.

 

[moephysics]

ok so Ro=0.2725Re and deo=3.85x10^8 m

so our equation would be somethin like GMe/Deo^2-GMo/0.2725Rer^2=0 ?

 

[Hootenanny]

ok so Ro=0.2725Re and deo=3.85x10^8 m

so our equation would be somethin like GMe/Deo^2-GMo/0.2725Rer^2=0 ?

Hmmm, not quite. Let's try it like this:

$$R_o + R_e = D_{eo}$$

Hence

$$R_o = D_{eo} - R_e$$

The using your equation

$$\frac{G M_o}{r_o^2} = \frac{G M_e}{r_e^2}$$

We obtain

$$\frac{M_o}{\left(D_{eo} - R_e\right)^2} = \frac{M_e}{R_e^2}$$

Do you follow?

 

[moephysics]

alright so you mean it that way ok so far so good...

 

[Hootenanny]

alright so you mean it that way ok so far so good...

Good. So to answer your question you need to know where the velocity of this rock can be considered zero. Therefore, you need to determine R_e.

Once you have R_e the problem is a straight forward application of conservation of energy.

 

[moephysics]

alright thank you very much , i don't know what i would've done without your help:shy:

 

[Hootenanny]

alright thank you very much , i don't know what i would've done without your help:shy:

No problem, it was a pleasure. Feel free to come back if you get stuck later on.

Merry Christmas!