Is the Book Right? Examining Conservation of Momentum

[phantomvommand]

Homework Statement

This is a problem from 200 Puzzling Physics Problems.
A student and ticket collector of mass m are in a stationary railway carriage of mass M. The collector realises the student has no ticket, so the student runs towards the end of the carriage, and the collector chases after him with speed v relative to the carriage. The student stops at the end of the carriage and jumps out. Find the velocity of the carriage when the collector stops at the open end of the carriage and watches the student escape.

Relevant Equations

Conservation of Momentum

My proposed solution:

When the student stops at the end, suppose the carriage is moving at speed u.
0 = (M+2m)u - m(v - u)
==> u = mv/ M+3m

After jumping out, the total momentum of the Carriage + collector system is 0 - mu = -m^2v/ M+3m.

By conservation of momentum for the Carriage + collector system, at the time when the collector stops at the open end:

-m^2v/ M+3m = 0 + (M+m)V, where V is the final velocity of the carriage + collector.

V = -m^2v/ (M+3m)(M+m)

The book (200 Puzzling Phys Problems) writes that the speed of the carriage when the student has reached the end and stopped (but not yet jumped) is u = mv/M+2m.

I suppose this is from the Conserve momentum equation: (M+2m)u = mv. Doesn't this forget the fact that v is a relative velocity?

Is the book right, and why am I wrong?

Thank you!

 

[BvU]

When the student stops at the end, suppose the carriage is moving at speed u.
0 = (M+2m)u - m(v - u)
==> u = mv/ M+3m

1) your lack of using brackets will cause errors -- if not now, then later
2) Your momentum balance should not have the ticket collector on both sides, but: ticket collector on one side and carriage + student on the other.

$\$

 

[phantomvommand]

1) your lack of using brackets will cause errors -- if not now, then later
2) Your momentum balance should not have the ticket collector on both sides, but: ticket collector on one side and carriage + student on the other.

$\$

I see my mistake now, thank you very much!