Confusing wording? Energy conservation part two

[Shazbot]

Homework Statement

A 900-kg two-stage rocket is traveling at a speed of 6.50 x 10^3 m/s away from Earth when a predesigned explosion separates the rocket into two sections of equal mass that then move with a relative speed (relative to each other) of 2.80 x 10^3 m/s along the original line of motion. a) What is the speed and direction of each segment after the explosion? b) How much energy was supplied by the explosion? [Hint: What is the change in kinetic energy as a result of the explosion?]

Homework Equations

E = K + U = (.5)mv^2 + mgy

The Attempt at a Solution

This doesn't look like too difficult a problem, but I'm confused by the wording that the text here uses. Specifically, the part where it says that their speed in relation to each other is 2.80 x 10^3 m/s which is slower than they were both moving a moment ago. If an explosion has separated the rocket into two pieces, wouldn't one be moving more quickly and the other be moving more slowly? Is it trying to say that one piece is now moving 1.40 x 10^3 m/s faster and the other is moving 1.40 x 10^3 m/s slower (this is all I could come up with)? Similarly, if they're both moving at that speed along the original line of motion, why does it ask for the direction of their motion as well? I don't see any indication in the problem that either piece has changed course... I'm sure it can't be too difficult a problem, but I really wish there had been some kind of accompanying diagram. If anyone could help by writing this out in a way I could understand it (or drawing it if it'd be easier) I'd be a happy camper.

 

[Doc Al]

This doesn't look like too difficult a problem, but I'm confused by the wording that the text here uses. Specifically, the part where it says that their speed in relation to each other is 2.80 x 10^3 m/s which is slower than they were both moving a moment ago.

They are talking about relative speed, not speed with respect to the ground. For example, if one is moving at 100 m/s up and the other at 101 m/s up, then their relative speed is 1 m/s.

If an explosion has separated the rocket into two pieces, wouldn't one be moving more quickly and the other be moving more slowly?

Yes.

The wording makes it sound as if they are both now moving at 2.80 x 10^3 m/s Similarly, if they're both moving at that speed along the original line of motion, why does it ask for the direction of their motion as well?

Direction can be up or down. I suggest you call up positive and down negative.

Hint: This in not an energy conservation problem. Mechanical energy is not conserved in an explosion. But something else is conserved. What?

 

[Shazbot]

Conservation of momentum, of course. I swear that's that I meant to say :tongue:

Wait, so then it's saying that one piece gets blown higher up and the other piece gets blown down (in relation to the original trajectory)? Wouldn't they then not be traveling "2.80 x 10^3 m/s along the original line of motion"? Is it something else? I'm having trouble visualizing what's happening here...

 

[Shazbot]

I guess my question could be written a bit more simply as: by "two sections of equal mass that then move with a relative speed (relative to each other) of 2.80 x 10^3 m/s along the original line of motion" does it mean that piece 1 is traveling at 2.80 x 10^3 m/s in relation to piece 2 (which would be traveling at 0 m/s in relation to piece 1)?

or

does it mean that both pieces are traveling at 2.80 x 10^3 m/s and, if so, how is this possible considering one has to be slowed down and the other sped up by the force of the explosion?

and

how does direction factor into all of this?

 

[Doc Al]

Conservation of momentum, of course. I swear that's that I meant to say

Good answer.

Wait, so then it's saying that one piece gets blown higher up and the other piece gets blown down (in relation to the original trajectory)? Wouldn't they then not be traveling "2.80 x 10^3 m/s along the original line of motion"? Is it something else? I'm having trouble visualizing what's happening here...

Originally, the two rocket pieces are traveling at the same speed: 6.50 x 10^3 m/s upwards. After the explosion, the top piece is traveling at speed Vt and the bottom piece is traveling at speed Vb. Their relative speed is 2.80 x 10^3 m/s. It's up to you, using conservation of momentum and the given data, to find the actual final speeds with respect to the ground.

I guess my question could be written a bit more simply as: by "two sections of equal mass that then move with a relative speed (relative to each other) of 2.80 x 10^3 m/s along the original line of motion" does it mean that piece 1 is traveling at 2.80 x 10^3 m/s in relation to piece 2 (which would be traveling at 0 m/s in relation to piece 1)?

Almost. The top piece is traveling at a relative speed of +2.80 x 10^3 m/s (up) with respect to the bottom piece and, conversely, the bottom piece is traveling at a relative speed of -2.80 x 10^3 m/s (down) with respect to the top piece. Note that I use + to mean "up" and - to mean "down".

does it mean that both pieces are traveling at 2.80 x 10^3 m/s and, if so, how is this possible considering one has to be slowed down and the other sped up by the force of the explosion?

It doesn't mean that.

how does direction factor into all of this?

By direction they mean sign. Is it going up or down? You know the top piece must be going up, since it picked up speed with the explosion. But the bottom piece may just slow down (and thus keep going up) or perhaps is now going in the opposite direction--down. That's part of what you need to figure out.

To solve this:
(1) Calling the final speeds Vt & Vb, write an equation for conservation of momentum.
(2) Write another equation for their relative velocity.