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A 285-g ball falls vertically downward, hitting the floor with a speed of \(2.5 \mathrm{m} / \mathrm{s}\) and rebounding upward with a speed of \(2.0 \mathrm{m} / \mathrm{s}\) (a) Find the magnitude of the change in the ball's momentum. (b) Find the change in the magnitude of the ball's momentum. (c) Which of the two quantities calculated in parts (a) and (b) is more directly related to the net force acting on the ball during its collision with the floor? Explain.

Short Answer

Expert verified
(a) 1.2825 kg m/s; (b) 0.1425 kg m/s; (c) Change in momentum is related to net force.

Step by step solution

01

Convert mass to kilograms

The mass of the ball is given in grams. To use it in calculations, convert the mass to kilograms by dividing by 1000. So, the mass of the ball is \( 285 \text{ g} = 0.285 \text{ kg} \).
02

Calculate initial momentum

The initial momentum of the ball when it hits the floor is calculated using the formula \( p = mv \), where \( m \) is the mass and \( v \) is the velocity. Since the ball is moving downward, the velocity is negative. Thus, the initial momentum is \( p_i = 0.285 \times (-2.5) \text{ kg m/s} = -0.7125 \text{ kg m/s} \).
03

Calculate final momentum

The final momentum is calculated when the ball rebounds upward at \( 2.0 \text{ m/s} \). The velocity here is positive as the direction is upward. Hence, the final momentum is \( p_f = 0.285 \times 2.0 \text{ kg m/s} = 0.57 \text{ kg m/s} \).
04

Calculate the magnitude of the change in momentum

The change in momentum is given by \( \Delta p = p_f - p_i \). Substitute the values to find: \[ \Delta p = 0.57 - (-0.7125) = 0.57 + 0.7125 = 1.2825 \text{ kg m/s} \]. Therefore, the magnitude of the change in momentum is \( 1.2825 \text{ kg m/s} \).
05

Calculate the change in the magnitude of momentum

The magnitude of the initial momentum is \(|-0.7125| = 0.7125 \text{ kg m/s}\), and the magnitude of the final momentum is \(0.57 \text{ kg m/s}\). The change in the magnitude is \(|p_f| - |p_i|\): \[ \Delta |p| = 0.57 - 0.7125 = -0.1425 \]. Taking the absolute value gives \(0.1425 \text{ kg m/s}\).
06

Identify relation to net force

The magnitude of the change in momentum \(1.2825 \text{ kg m/s}\) is used to calculate the net force acting on the ball because according to Newton's second law, force is related to the change in momentum over time. The change in momentum (not the change in the magnitude) is more directly related to the net force.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Change in Momentum
Momentum is a measurement of mass in motion. It is the product of an object's mass and velocity. When a force acts on an object, its momentum changes. The concept of change in momentum is significant when analyzing how objects in motion respond to forces.

In the scenario of a rebounding ball, as it hits the floor, it is first moving downward, then rebounds upward. This change in direction and speed signifies a change in its momentum.

The change in momentum \( \Delta p \) is calculated using the formula: \[ \Delta p = p_f - p_i \] where \( p_f \) is the final momentum and \( p_i \) is the initial momentum.

For the ball, this change was computed to be \( 1.2825 \text{ kg m/s} \), indicating a significant shift as it bounced up from the floor.
Collision
A collision occurs when two or more objects hit each other. In physics, the study of collisions often focuses on how the motion of objects changes due to their impact.

In this exercise, the collision involves a ball and the floor. The ball strikes the floor, momentarily stopping before springing back upward. During this brief interaction, force is exchanged between the ball and the floor, altering the ball's momentum.

Collisions can be described in terms of momentum conservation, kinetic energy changes, and impact forces. Understanding these properties helps students explain why objects behave as they do when they collide.

The example under discussion shows an inelastic-type collision mainly due to the loss in speed when the ball rebounds, highlighting that some kinetic energy is transformed to other forms like sound or heat.
Net Force
The concept of net force revolves around the idea that various forces can act on an object simultaneously. The net force is the vector sum of all forces acting on an object, determining its acceleration and subsequent motion.

In this case, as the ball hits the floor and bounces back, it experiences a normal force from the ground countering gravitational pull. This interchange results in a change in motion, analogous to a net force acting during the collision interval.

This net force then directly influences the change in momentum of the ball due to its interaction with the floor, allowing the ball to reverse direction. Such force predominantly affects the ball over only the brief time of impact.
Newton's Second Law
Newton's Second Law is a fundamental principle that states the force acting on an object is equal to the mass of the object times its acceleration \( F = ma \). It offers a direct insight into how forces affect motion.

An extension of this law relates to momentum, expressed as \( F = \frac{\Delta p}{\Delta t} \), where \( \Delta p \) is the change in momentum and \( \Delta t \) is the change in time. This relationship helps explain how forces applied over time cause changes in an object's momentum.

In this case, the force exerted by the floor on the ball during the collision not only changes the direction of the ball but also showcases how momentum change relates directly to the net force. By using Newton's Second Law, we can better understand the forces at play and their effects on the ball's movement.

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