Problem 72 - University Physics with Modern Physics

Question

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 200 \(\mathrm{kg}\) and is traveling east with a velocity of magnitude 5.00 \(\mathrm{m} / \mathrm{s} .\) Find the final velocity of the car in each case, assuming that the handcar does not leave the tracks. (a) A \(25.0-\mathrm{kg}\) mass is thrown sideways out of the car with a velocity of magnitude 2.00 \(\mathrm{m} / \mathrm{s}\) relative to the car's initial velocity. (b) A \(25.0-\mathrm{kg}\) mass is thrown backward out of the car with a velocity of 5.00 \(\mathrm{m} / \mathrm{s}\) relative to the initial motion of the car. (c) A 25.0 -kg mass is thrown into the car with a velocity of 6.00 \(\mathrm{m} / \mathrm{s}\) relative to the ground and opposite in direction to the initial velocity of the car.

Step-by-Step Solution

Answer

(a) 5.00 m/s, (b) 7.14 m/s, (c) 3.78 m/s.

Step 1: Understand the Problem

We are tasked with finding the final velocity of a handcar in three different scenarios where masses are being thrown in different directions. The initial momentum of the system must be adjusted according to the momentum changes in each case.

Step 2: Calculate Initial Momentum

The initial momentum of the system can be calculated using the formula \( p = mv \), where \( m \) is the total mass (200 kg) and \( v \) is the velocity (5.00 m/s). Thus, the initial momentum is \( p_i = 200 \, ext{kg} \times 5.00 \, ext{m/s} = 1000 \, ext{kg} \cdot \, ext{m/s} \).

Step 3a: Scenario (a) - Mass Thrown Sideways

In this scenario, since the mass is thrown sideways, it will not affect the forward momentum calculated along the direction of motion. Thus, the mass thrown out will not change the momentum in the east direction.

Step 4a: Calculate Final Velocity for Scenario (a)

The effective mass of the car remains 200 kg, and since no momentum is lost in the east direction, final velocity \( v_{f} \) is 5.00 m/s.

Step 3b: Scenario (b) - Mass Thrown Backward

The 25.0 kg mass is thrown backward at 5.00 m/s relative to the car. The rearward velocity of the mass in the ground frame of reference will be 5.00 m/s (initial car speed) + 5.00 m/s = 10.00 m/s.

Step 4b: Calculate Final Velocity for Scenario (b)

Use the conservation of momentum: \( p_i = p_f \), where \( p_f = (200 \, ext{kg} - 25.0 \, ext{kg})v_{f}' + (25.0 \, ext{kg}) \times (-10.00 \, ext{m/s}) \). Calculate the forward velocity: \( 1000 = 175v_{f}' - 250 \). Solving gives \( v_{f}' = 7.14 \, ext{m/s} \).

Step 3c: Scenario (c) - Mass Thrown into the Car

A 25.0 kg mass moves opposite to the car's velocity at 6.00 m/s. Calculate its momentum: \( 25.0 \, ext{kg} \times (-6.00) \, ext{m/s} = -150 \, ext{kg} \cdot \, ext{m/s} \).

Step 4c: Calculate Final Velocity for Scenario (c)

Using the conservation of momentum: \( p_i + p_m = p_f \), where \( p_f = (200 \, ext{kg} + 25.0 \, ext{kg})v_{f}'' \). Therefore, \( 1000 - 150 = 225v_{f}'' \). Solving yields \( v_{f}'' = 3.78 \, ext{m/s} \).

Key Concepts

Momentum

Momentum is a fundamental concept in physics that describes the motion of an object. It is the product of an object's mass and its velocity. Therefore, the formula for momentum is given by \( p = mv \). Momentum is a vector quantity, which means it has both magnitude and direction.
Momentum helps us understand how objects behave when they collide or interact with each other. The principle of conservation of momentum states that the total momentum of a closed system remains constant, provided no external forces act on it.
In the railroad handcar exercise, momentum conservation is crucial. The initial momentum of the handcar and any objects on it must equal the total momentum after any interactions, like masses being thrown out or into the car.

Velocity

Velocity is a key physics concept that refers to the speed of an object in a given direction. Unlike speed, which is scalar, velocity is a vector and has both magnitude and direction. In formulas, velocity is typically denoted as \( v \), and its unit is meters per second (\( m/s \)).
In many problems, including the railroad handcar scenario, understanding how velocities change due to interactions like collisions or separations is essential. For example:

• When a mass is thrown backward from the handcar, its velocity relative to the ground and the car changes.
• The velocity's direction also influences the momentum calculations.

This means that when calculating final conditions, like in the handcar exercise, we must carefully track the direction and magnitude of velocities involved.

Railroad Handcar

A railroad handcar is a simple rail vehicle that operates on tracks and is often used as an example in physics problems to study motion and forces. These exercises usually assume frictionless conditions and negligible air resistance to focus solely on the principles of momentum and motion.
In practical terms, a handcar is depicted moving along a straight track, in this case, with an initial mass of 200 kg and a velocity of 5.00 m/s. The challenge is to find the final velocity after different scenarios of mass ejection or addition.
Such exercises illustrate conservation principles, showing how actions like throwing a mass off or into the handcar affect the overall movement. Simplifying assumptions help students grasp the concepts behind real-world physics through controlled scenarios.

Mass and Motion

Mass and motion are intimately connected in physics. Mass refers to the amount of matter in an object, and it plays a central role in defining momentum, as seen in the equation \( p = mv \).
Motion refers to the change in the position of an object over time, which can be linear or rotational. In this exercise, we're focused on linear motion along the track.
When changes occur, such as a mass being thrown off or added to the handcar, both the mass of the system and its velocity change, affecting the overall momentum:

• Throwing mass backward increases velocity due to a decrease in mass and momentum conservation.
• Adding mass decreases velocity as the total mass increases.

Understanding these dynamics is crucial in physics to predict system behavior accurately.