A railroad handcar is moving along straight, frictionless...

A railroad handcar is moving along straight, frictionless tracks with negligible air resistance. In the following cases, the car initially has a total mass (car and contents) of $150 \mathrm{~kg}$ and is traveling east with a velocity of magnitude $4.70 \mathrm{~m} / \mathrm{s}$. Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) $\mathrm{A}$ $20.0 \mathrm{~kg}$ mass is thrown sideways out of the car with a velocity of magnitude $2.40 \mathrm{~m} / \mathrm{s}$ relative to the car's initial velocity. (b) A $20.0 \mathrm{~kg}$ mass is thrown backward out of the car with a velocity of $4.70 \mathrm{~m} / \mathrm{s}$ relative to the initial motion of the car. (c) A $20.0 \mathrm{~kg}$ mass is thrown into the car with a velocity of $6.30 \mathrm{~m} / \mathrm{s}$ relative to the ground and opposite in direction to the initial velocity of the car.

A railroad handcar is moving along straight, frictionless tracks with negligible air resistance. In the following cases, the car initially has a total mass (car and contents) of $150 \mathrm{~kg}$ and is traveling east with a velocity of magnitude $4.70 \mathrm{~m} / \mathrm{s}$. Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks.
(a) $\mathrm{A}$ $20.0 \mathrm{~kg}$ mass is thrown sideways out of the car with a velocity of magnitude $2.40 \mathrm{~m} / \mathrm{s}$ relative to the car's initial velocity. (b) A $20.0 \mathrm{~kg}$ mass is thrown backward out of the car with a velocity of $4.70 \mathrm{~m} / \mathrm{s}$ relative to the initial motion of the car. (c) A $20.0 \mathrm{~kg}$ mass is thrown into the car with a velocity of $6.30 \mathrm{~m} / \mathrm{s}$ relative to the ground and opposite in direction to the initial velocity of the car.

Key Concepts

Vector Decomposition

When objects are thrown in directions that are not aligned with the initial motion (for example, sideways or at an angle), it becomes necessary to decompose velocities into their respective components. This allows for the proper application of conservation laws in each direction, ensuring that momentum is conserved not only in magnitude but also in direction.

System Mass Change and Isolation

In problems where portions of a system are ejected or added, it is important to correctly identify the system boundaries. The mass removal from the handcar alters the system’s mass, and the subsequent momentum calculations must account for this change while conserving the overall momentum of the isolated system.

Conservation of Linear Momentum

This principle states that if no external forces are acting on a system, the total momentum of the system remains constant before and after an event. In the context of the problem, as the mass is thrown out of the moving handcar, the total momentum of the combined system (handcar plus thrown mass) must be conserved, which sets up the equations needed to compute the final velocity of the handcar after the event.

Relative Velocity and Frame of Reference

Understanding relative velocity is crucial because the speeds at which objects are thrown are given with respect to the handcar or the ground. Transformations between different frames of reference, such as converting a velocity measured relative to the car into a velocity relative to the ground, are essential for correctly applying momentum conservation laws.

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