Particle Shot Tangentially to Surface

[trevor51590]

Homework Statement

A particle mass m is shot tangentially to the surface of a planet mass M radius R₀ at 3/4 escape velocity. Determine the maximum radius the particle achieves from the center of the planet

Homework Equations

Escape Velocity : √(2GM/r)
Energy due to Gravity : -GMm/r²
F=ma
Angular Force : F=mv²/r
Conservation of Angular Momentum : L = r x p or L=mvr

The Attempt at a Solution

Honestly - the beginning of the attempt is where I'm having the issue = I can't decide if I want to use conservation of energy, conservation of angular momentum.

My thought process tells me alright, the particle is shot at a velocity v₀ (3/4 escape velocity). This particle has a force acting upon it - the force due to gravity. I can picture exactly what happens to the particle. it won't be shot out fast enough to reach orbit, it reaches a maximum radius r, and then in a parabolic trajectory falls back down to the planet.

Quite honestly - I feel like it should be a very simple solution, I just can't wrap my head around it. Perhaps a tip to start will give me the momentum to find the solution.

Thanks!

 

[vela]

Homework Statement

A particle mass m is shot tangentially to the surface of a planet mass M radius R₀ at 3/4 escape velocity. Determine the maximum radius the particle achieves from the center of the planet

Homework Equations

Escape Velocity : √(2GM/r)
Energy due to Gravity : -GMm/r²
F=ma
Angular Force : F=mv²/r

Centripetal force, not angular force (whatever that means).

Conservation of Angular Momentum : L = r x p or L=mvr

L=mvr sin θ

The Attempt at a Solution

Honestly - the beginning of the attempt is where I'm having the issue = I can't decide if I want to use conservation of energy, conservation of angular momentum.

You don't need to choose. You have to use both.

My thought process tells me alright, the particle is shot at a velocity v₀ (3/4 escape velocity). This particle has a force acting upon it - the force due to gravity. I can picture exactly what happens to the particle. it won't be shot out fast enough to reach orbit, it reaches a maximum radius r, and then in a parabolic trajectory falls back down to the planet.

Elliptical trajectory, not parabolic.

 

[trevor51590]

Thank you!

I used conservation of energy

1/2mv_o²-C/A=1/2mv_f²-C/A

For v_f I used conservation of angular momentum to put it into terms of v_i

Thanks again!