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.

 

[Nickie]

However, in the bug's frame, the momentum is simply the negative of the truck's momentum, whereas in the truck's frame, the momentum is the negative of the bug's momentum plus the truck's momentum. This is because the truck's frame is moving and has its own momentum, while the bug's frame is stationary and does not have its own momentum. Therefore, the transformation of m_B\boldsymbol v_{BT} is not necessary and does not change the overall momentum of the system. In conclusion, there is nothing wrong with the calculations and the difference in magnitude of the momenta is simply due to the different reference frames.