Conservation of momentum and relative velocity

[anotherghost]

Homework Statement

So you've got two people standing on a boat, everything at rest, and the masses of both people as well as the boat are known. Let's call them Richard and Sandra, so the variables are Mr (mass of Richard), Ms (mass of Sandra), and Mb (mass of boat). When one of these people jumps off the side of the boat, they have a velocity of 3.5m/s relative to the boat.

The problem asks to find the final speed of the boat Mbf after:
a) Both people jump off in the same direction at the same time.
b) First Richard jumps off, then Sandra a few seconds later, both in the same direction.
c) First Sandra jumps off, then Richard a few seconds later, both in the same direction.
d) First Richard jumps off to the right, then a few seconds later Sandra jumps off to the left.

Homework Equations

I guess just conservation of momentum.

The Attempt at a Solution

a) 0 = Mb * Vbf + (Mr + Ms)(3.5 + Vbf)

b) 0 = (Mb + Ms) * Vbs + Mr * (3.5 + Vbs)
(Mb + Ms) * Vbs = Mb * Vbf + Ms * (3.5 + Vbf)

c) 0 = (Mb + Mr) * Vbr + Ms * (3.5 + Vbr)
(Mb + Mr) * Vbr = Mb * Vbf + Mr * (3.5 + Vbf)

d) 0 = (Mb + Ms) * Vbs + Mr * (3.5 + Vbs)
(Mb + Ms) * Vbs = Mb * Vbf + Ms * (-3.5 + Vbs)

After these equations, of course, you solve for Vbf. I can do that part easily but I want to make sure the physics of these equations are right. My rationale is that I can first equate 0 (because the system starts at rest) with the first jump, and then equate the system without the person who jumped off with the second jump. I solve the first equation for the velocity needed in the left side of the second equation, then I solve the second equation for Vbf.

I am not sure if these are even remotely right or not - they make sense to me, but I'm not sure! Relative velocity is confusing. Please somebody give me a hand, even if it's only on one part.

Thanks.

 

[Delphi51]

Looks good to me. Very tricky business.

 

[kerimek]

Your approach to solving these problems using conservation of momentum and relative velocity seems correct. In all four scenarios, you are considering the initial momentum of the system (which is zero since everything is at rest) and the final momentum of the system (after the person jumps off) and equating them. This is a valid approach and will give you the correct solutions.

One thing to note is that in scenario d), the velocity of Sandra should be negative since she is jumping off in the opposite direction of Richard. So the equation for d) should be (Mb + Ms) * Vbs = Mb * Vbf + Ms * (-3.5 + Vbs). Other than that, your equations and approach seem correct.

In general, when solving problems involving conservation of momentum and relative velocity, it is important to clearly define your coordinate system and velocities to avoid any confusion. Overall, you have a good understanding of these concepts and your approach to solving these problems is sound. Keep up the good work!