Spaceship Escape speed understanding

[Patdon10]

Homework Statement

A spaceship is launched from the Earth's surface with a speed v. The radius of the Earth is R. What will its speed be when it is very far from the Earth? (Use any variable or symbol stated above along with the following as necessary: G for the gravitational constant, m for the mass of the spaceship, and M for the mass of the Earth.)

What is V_f?


2. The attempt at a solution

I'm really not sure what my teacher is looking for. I tried sqroot(2GM/R) to no avail. Any ideas?

I also know vf = vi + at. What could I substitute in for the acceleration?

 

[collinsmark]

I also know vf = vi + at. What could I substitute in for the acceleration?

The v_f - v_i = at formula only works for uniform acceleration. But that doesn't apply here because the acceleration changes throughout the trip.

And by the way, I'm pretty sure you're supposed to assume that the spaceship was launched into space as if launched from a catapult. In other words, once it gets away from the Earth's surface, it drifts naturally without using any significant thrust or propulsion. Assume the ship uses up all of its propulsion/fuel/etc, when launching, immediately after which the spaceship has mass m and speed v.

Here are some questions to ask yourself.

• What is the initial kinetic energy of the spaceship?
• What is the gravitational potential of Earth (with respect to infinity)?
• Knowing Earth's gravitational potential, what is the difference in gravitational potential energy between a spaceship of mass m on the surface of the Earth and infinity?
• So after reaching an infinite distance from Earth (well, let's just say "a long way from Earth"), how much kinetic energy is left over?

 

[Patdon10]

The equation I'm getting is total energy = (1/2)mv^2 - G*(mM/R+h)

How could I solve for v final from having the total energy? h = infinity

 

[collinsmark]

The equation I'm getting is total energy = (1/2)mv^2 - G*(mM/R)

'Looks good but just make sure you note that the v in the above equation is the initial speed, v_i.

K.E._final = ½mv_i² - G(Mm/R)​

How could I solve for v final from having the total energy?

Ummm, I'm sure you already know this,

K.E._final = ½mv_f²​

 

[Patdon10]

well, yeah. I know that final kinetic energy is equal to that. However, why is it kinetic energy? Isn't it total Mechanical energy?...

You know what, you're right. Isn't it only kinetic energy because the potential energy is 0 (it's very far away)?

 

[Patdon10]

I tried

Still telling me it's the wrong answer : /

 

[collinsmark]

I tried

Still telling me it's the wrong answer : /

I don't see anything wrong with your answer. But the program might be expecting you to simplify a little more maybe. You could multiply the 2/m through the other terms, getting rid of the variable m in the process.

 

[Patdon10]

Alright. I'll give it a try. Thanks a lot for the speedy responses.