Kinematics, work-energy, conservation, more, less?

[barrylwires]

Homework Statement

Rocket launched runs out of fuel after accelerating "for a short distance," coasts to a maximum altitude, then falls back. Known: rocket mass, max. altitude, g. Find velocity when rocket runs out of fuel.

Homework Equations

There doesn't seem to be enough information to use kinematic eqs: no time for anything, no initial acceleration; only maximum height. We know the rocket force must be at least mg, but it's clearly more, since it coasts upwards.

The Attempt at a Solution

Seems to need energy methods, but I can't see the way. E=K+U, but E=U (at top) doesn't equal E at bottom, where K and U are both zero. And what can we know about K+U at y_fuelanyway? Do I need work, force? Do I need W_{nonconservative}=∆K+∆U? Does momentum enter this? I know that force changes from rocket to gravity at y_fuel, but I'm not sure what to do with that. I could use a shove in the right direction.

If I try K₁+U₁=K₂+U₂ at y₁=no fuel and y₂=max. altitude, I just wind up with v²=2g(y₂-y₁), the kinematic eq., and I have two unknowns. I need another expression with one of these variables in it. Momentum? Kinematics? Thanks.

 

[LowlyPion]

Homework Statement

Rocket launched runs out of fuel after accelerating "for a short distance," coasts to a maximum altitude, then falls back. Known: rocket mass, max. altitude, g. Find velocity when rocket runs out of fuel.

Homework Equations

There doesn't seem to be enough information to use kinematic eqs: no time for anything, no initial acceleration; only maximum height. We know the rocket force must be at least mg, but it's clearly more, since it coasts upwards.

The Attempt at a Solution

Seems to need energy methods, but I can't see the way. E=K+U, but E=U (at top) doesn't equal E at bottom, where K and U are both zero. And what can we know about K+U at y_fuelanyway? Do I need work, force? Do I need W_{nonconservative}=∆K+∆U? Does momentum enter this? I know that force changes from rocket to gravity at y_fuel, but I'm not sure what to do with that. I could use a shove in the right direction.

If I try K₁+U₁=K₂+U₂ at y₁=no fuel and y₂=max. altitude, I just wind up with v²=2g(y₂-y₁), the kinematic eq., and I have two unknowns. I need another expression with one of these variables in it. Momentum? Kinematics? Thanks.

Welcome to PF.

If you know the max altitude you know a lot don't you?

Vi² = 2*g*x

since you know the max altitude is where vf = 0

The only thing off is the "very short distance".

 

[barrylwires]

Hello and thank you. Yes. y₂ = y_max does tell me a lot, but I don't see what resources I have to determine either a) the distance y₁ is above the ground = y₀ or b) what time the rocket ran out of fuel. I do see that v₂ and v₀= 0 and that v₁² = -2g(y2-y1). Are you saying there is a way to determine y₁?

 

[LowlyPion]

Hello and thank you. Yes. y₂ = y_max does tell me a lot, but I don't see what resources I have to determine either a) the distance y₁ is above the ground = y₀ or b) what time the rocket ran out of fuel. I do see that v₂ and v₀= 0 and that v₁² = -2g(y2-y1). Are you saying there is a way to determine y₁?

The problem statement says short distance.

If you have Δy << y, and v² = 2*g*y

then I'd say (y/(y-Δy))^1/2 ≈ 1

 

[barrylwires]

Great. Got it.
When the problem said "Assume it was not far from the surface [of the moon] at that time [when it ran out of fuel]," I thought the point might have been to use mg_MOON, rather than Gm_Rm_M/r², but I'm happy to know that there isn't a way to find an actual value from the information given.
Many thanks!