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Chapter 9: Problem 18

A \(0.70 \mathrm{~kg}\) ball moving horizontally at \(5.0 \mathrm{~m} / \mathrm{s}\) strikes a vertical wall and rebounds with speed \(2.0 \mathrm{~m} / \mathrm{s}\). What is the magnitude of the change in its linear momentum?

Short Answer

Expert verified
The magnitude of the change in linear momentum is 4.9 kg m/s.

Step by step solution

01

Understand Initial Momentum

The initial momentum of an object is given by the product of its mass and velocity. For the ball moving horizontally, the initial momentum is \( p_i = m \times v_i \), where \( m = 0.70 \text{ kg} \) and \( v_i = 5.0 \text{ m/s} \). Thus, \( p_i = 0.70 \times 5.0 \).
02

Calculate Initial Momentum

Calculate the initial momentum using the formula from Step 1. \[ p_i = 0.70 \times 5.0 = 3.5 \text{ kg m/s}. \]
03

Understand Final Momentum

After striking the wall, the ball rebounds in the opposite direction with a velocity of \( v_f = -2.0 \text{ m/s} \). The final momentum is given by \( p_f = m \times v_f \), where \( v_f = -2.0 \text{ m/s} \).
04

Calculate Final Momentum

Calculate the final momentum. Note that the velocity is negative because the ball rebounds in the opposite direction.\[ p_f = 0.70 \times (-2.0) = -1.4 \text{ kg m/s}. \]
05

Determine Change in Momentum

The change in momentum \( \Delta p \) is the difference between the final and initial moments. This is given by \( \Delta p = p_f - p_i \).
06

Calculate Magnitude of Change in Momentum

Substitute the values of initial and final momentum into the change in momentum formula and calculate.\[ \Delta p = (-1.4) - (3.5) = -4.9 \text{ kg m/s}. \]The magnitude is the absolute value: \(|\Delta p| = 4.9 \text{ kg m/s}.\)

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Key Concepts

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

Change in Momentum
Change in momentum refers to the difference between the momentum an object has before and after an event, such as a collision or a bounce. It quantifies how much an object's motion has altered. For a straightforward understanding, momentum can be thought of as 'quantity of motion'.
  • Mathematically, change in momentum is represented as \( \Delta p = p_f - p_i \), where \( p_f \) is the final momentum and \( p_i \) is initial momentum.
  • In this exercise, the ball collided with the wall, reversing its direction, heavily affecting its momentum.
  • To find the magnitude of the change, calculate \( \Delta p \) and take the absolute value. Here, this results in \( |\Delta p| = 4.9 \text{ kg m/s} \).
Momentum change is a crucial concept in physics, relating closely to forces acting upon the masses over time according to Newton鈥檚 second law, emphasizing the dynamics of interactions.
Initial Momentum
The initial momentum of an object is simply the momentum it carries before an interaction or change in motion occurs. It's calculated by multiplying the object's mass by its velocity at the initial phase.
  • For example, in our exercise, the ball initially moves at \( 5.0 \text{ m/s} \), with a mass of \( 0.70 \text{ kg} \).
  • Its initial momentum \( p_i \) can be calculated as: \( p_i = m \times v_i = 0.70 \times 5.0 = 3.5 \text{ kg m/s} \).
Initial momentum is a foundational concept in physics calculations, as it provides the starting point from which changes due to external forces are measured, establishing a benchmark for analyzing results and effects.
Final Momentum
Final momentum is the momentum of the object after undergoing an event or experiencing a force, often differing from the initial momentum.
  • In the problem, after hitting the wall, the ball reverses direction, which is critical here as it affects velocity direction, making it negative.
  • The ball then has a velocity of \(-2.0 \text{ m/s}\), keeping the mass as \(0.70 \text{ kg}\).
  • Therefore, its final momentum \( p_f \) is \( p_f = m \times v_f = 0.70 \times (-2.0) = -1.4 \text{ kg m/s} \).
Understanding final momentum helps describe how external actions, like collisions, change the sate of motion. It's integral for determining results like momentum conservation or the force exerted on objects.

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Most popular questions from this chapter

Two particles \(P\) and \(Q\) are released from rest \(1.0 \mathrm{~m}\) apart. \(P\) has a mass of \(0.10 \mathrm{~kg}\), and \(Q\) a mass of \(0.30 \mathrm{~kg} . P\) and \(Q\) attract each other with a constant force of \(1.0 \times 10^{-2} \mathrm{~N}\). No external forces act on the system. (a) What is the speed of the center of mass of \(P\) and \(Q\) when the separation is \(0.50 \mathrm{~m}\) ? (b) At what distance from \(P\) 's original position do the particles collide?

In a game of pool, the cue ball strikes another ball of the same mass and initially at rest. After the collision, the cue ball moves at \(3.50 \mathrm{~m} / \mathrm{s}\) along a line making an angle of \(22.0^{\circ}\) with the cue ball's original direction of motion, and the second ball has a speed of \(2.00 \mathrm{~m} / \mathrm{s}\). Find (a) the angle between the direction of motion of the second ball and the original direction of motion of the cue ball and (b) the original speed of the cue ball. (c) Is kinetic energy (of the centers of mass, don't consider the rotation) conserved?

Two blocks of masses \(1.0 \mathrm{~kg}\) and \(3.0 \mathrm{~kg}\) are connected by a spring and rest on a frictionless surface. They are given velocities toward each other such that the \(1.0 \mathrm{~kg}\) block travels initially at \(1.7 \mathrm{~m} / \mathrm{s}\) toward the center of mass, which remains at rest. What is the initial speed of the other block?

A \(1.2 \mathrm{~kg}\) ball drops vertically onto a floor, hitting with a speed of \(25 \mathrm{~m} / \mathrm{s} .\) It rebounds with an initial speed of \(10 \mathrm{~m} / \mathrm{s}\). (a) What impulse acts on the ball during the contact? (b) If the ball is in contact with the floor for \(0.020 \mathrm{~s}\), what is the magnitude of the average force on the floor from the ball?

Ball \(B\), moving in the positive direction of an \(x\) axis at speed \(v\), collides with stationary ball \(A\) at the origin. \(A\) and \(B\) have different masses. After the collision, \(B\) moves in the negative direction of the \(y\) axis at speed \(v / 2 .\) (a) In what direction does \(A\) move? (b) Show that the speed of \(A\) cannot be determined from the given information.
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