Question

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

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

Step-by-step solution

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.

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

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

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

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.

Key concepts

Momentum

Momentum is a fundamental concept in physics that describes the motion of an object. It's a vector quantity, meaning it has both magnitude and direction. The formula for momentum is given by:

• Momentum (p) = mass (m) x velocity (v)

To fully understand momentum, consider that it reflects the tendency of an object to keep moving. The more mass an object has, or the faster it moves, the higher its momentum. In the context of our problem, initially, the entire system (student, cart, and ball) is at rest. Therefore, the total momentum of the system is 0. This means any action taken within the system, like throwing the ball, must abide by the law of conservation of momentum. The momentum gained by the ball must be equal and opposite to the momentum gained by the student-cart system. This keeps the total system momentum at 0, despite the internal changes.

Velocity

Velocity is a crucial part of understanding motion in physics. It describes how fast and in which direction an object moves. Unlike speed, velocity is a vector quantity, which means direction matters. In solving physics problems, especially those involving momentum, it's important to correctly apply velocity in equations. Here, the student throws the ball at a velocity of 10 m/s relative to themselves. With this information, we then solve for the student's velocity relative to the ground over the cart. Both velocities must be considered when determining the ball’s velocity with respect to the earth. In this problem, by considering the velocity of the student-cart system (which moves backward at -1 m/s) and adding it to the ball's throwing velocity (10 m/s), we conclude that the ground-relative velocity of the ball is 9 m/s. Understanding velocity and how it interacts in different reference frames helps unravel these types of physics problems efficiently.

Physics Problem Solving

Physics problem solving involves breaking down complex problems into manageable steps, using principles and formulas effectively. In the exercise given, we used the law of conservation of momentum. Here's the step-by-step approach to solving this type of problem:

• Identify the System: Here, the system includes the student, cart, and the ball.
• Analyze Momentum: Start by calculating the initial total momentum. Note that initially being at rest indicates zero total momentum.
• Apply Conservation Laws: When the ball is thrown, the momentum principle helps find the resulting movements within the system.
• Relate Velocities: Use momentum to find the new velocities of each component in the system and ensure the sum is consistent with initial conditions.

By systematically applying these steps, students can tackle physics problems involving systems, velocities, and forces, building up their problem-solving skills in the process. The ability to break down complex motion into parts governed by physics principles like momentum and velocity is crucial for any aspiring physicist or engineer.