A bug of inertia \(m_{\mathrm{B}}\) collides with the windshield of a Mack truck of inertia \(m_{\mathrm{T}} \gg m_{\mathrm{B}}\) at an instant when the relative speed of the two is \(v_{\mathrm{BT}}\).

(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 \(m_{\mathrm{B}} v_{\mathrm{BT}}\) 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 \(m_{\mathrm{T}} v_{\mathrm{BT}}\) from the expression.

(c) Is there something wrong here? How can we change the momentum by a small amount \(m_{\mathrm{B}} v_{\mathrm{BT}}\) doing the transformation one way and by a large amount \(m_{\mathrm{T}} v_{\mathrm{BT}}\) doing the transformation the other way?