Transforming momentum between inertial reference frames

[bkraabel]

Homework Statement

A bug of inertia m_B collides with the windshield of a Mack truck of inertia m_T \gg m_B at an instant when the relative velocity of the two is \boldsymbol v_{BT}.
(a) Express the system momentum in the truck’s reference frame, then transform that expression
to the bug’s reference frame, and in so doing remove m_B\boldsymbol v_{BT} from the expression. (Remember, in the bug’s reference frame, the bug is initially at rest and the truck is moving.)
(b) Now express the system momentum in the bug’s reference frame, then transform that expression to the truck’s reference frame, and in so doing remove m_T\boldsymbol v_{BT} from the expression.
(c) Is there something wrong here? How can we change the momentum by a small amount m_Bv_{BT} doing the transformation one way and by a large amount m_Tv_{BT} doing the transformation the other way?

Homework Equations

Take the bug's direction as the positive direction. System momentum in bug frame is
\boldsymbol p_{sys,B}=-m_T\boldsymbol v_{BT}
System momentum in truck frame is
\boldsymbol p_{sys,T}=m_B\boldsymbol v_{BT}

The Attempt at a Solution

I can see that the magnitude of the momentum is much larger in the bug frame, but I don't get the part about removing m_B\boldsymbol v_{BT}. It doesn't seem necessary or even possible. I understand that the absolute magnitude of the momentum in different inertial reference frames is not important. What is important is the difference between momenta in two inertial frames. This difference should be the same in the two frames.

 

[Simon Bridge]

inertial of truck = M, inertial of bug = m.
relative velocity is v.

In truck frame, p = mv
In bug frame, P = -Mv <---<<< written by inspection.
But is there a rule for transforming momentum from one frame to another that you have been given in your course notes?

http://en.wikipedia.org/wiki/Galilean_transformation
https://www.physicsforums.com/showthread.php?t=108631

 

[bkraabel]

No special rules given for momentum transformation, just the regular Galilean transformation rules for transforming velocity between different frames. But applying a Galilean transformation just gives you the equations we've already written above.

 

[Simon Bridge]

Well yes - but you asked about the process.
You can just write down the equation by inspection because, basically, you know the outcome in advance... or you can start with one and formally apply the transformation step-by-step and demonstrate you get the same thing.