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Chapter 8: Problem 57

A student with a mass of \(40.0 \mathrm{~kg}\) can throw a \(5.00-\mathrm{kg}\) ball with a relative speed of \(10.0 \mathrm{~m} / \mathrm{s}\). The student is standing at rest on a cart of mass \(10.0 \mathrm{~kg}\) that can move without friction. If the student throws the ball horizontally, what will the velocity of the ball with respect to the ground be?

Short Answer

Expert verified
Answer: The velocity of the ball with respect to the ground will be 9 m/s.

Step by step solution

01

Calculate the initial total momentum of the system

Initially, both the student-cart system and the ball are at rest. Therefore, the initial momentum of the system is 0.
02

Calculate the momentum of the ball after being thrown

The student throws the ball with a relative speed of 10 m/s. To find the momentum of the ball after being thrown, we need to multiply its mass by its relative speed: Momentum = mass x velocity Momentum of ball = 5 kg x 10 m/s = 50 kgm/s
03

Apply conservation of momentum

The conservation of momentum principle states that the total momentum of an isolated system remains constant if no external forces act on the system. In this case, the initial total momentum was 0, so the final total momentum must be 0 as well. Therefore, the momentum of the student-cart system after the student throws the ball must be equal in magnitude and opposite in direction to the momentum of the ball: Momentum of student-cart system = - Momentum of ball Momentum of student-cart system = - 50 kgm/s
04

Calculate the velocity of the student-cart system

To find the velocity of the student-cart system after the student throws the ball, we need to divide the momentum of the student-cart system by the total mass of the student and the cart: Velocity = Momentum / Mass Velocity of student-cart system = -50 kgm/s / (40 kg + 10 kg) Velocity of student-cart system = -50 kgm/s / 50 kg = -1 m/s
05

Calculate the velocity of the ball with respect to the ground

Now that we have the velocity of the student-cart system and the ball with respect to the student, we can calculate the velocity of the ball with respect to the ground by adding the two velocities: Velocity of ball with respect to ground = Velocity of ball (relative to student) + Velocity of student-cart system Velocity of ball with respect to ground = 10 m/s + (-1 m/s) = 9 m/s The velocity of the ball with respect to the ground will be 9 m/s.

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Most popular questions from this chapter

Find the following center-of-mass information about objects in the Solar System. You can look up the necessary data on the Internet or in the tables in Chapter 12 of this book. Assume spherically symmetrical mass distributions for all objects under consideration. a) Determine the distance from the center of mass of the Earth-Moon system to the geometric center of Earth. b) Determine the distance from the center of mass of the Sun-Jupiter system to the geometric center of the Sun.

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