Question

A 48.95 -kg boy is plaving rollerblade dodgeball. His rollerblades are frictionless, and he is initially at rest. A dodgeball with a mass of \(511.1 \mathrm{~g}\) is thrown directly at him with a speed of \(23.63 \mathrm{~m} / \mathrm{s} .\) He catches the dodgeball. With what speed does the boy move after the catch?

Short answer

Answer: To find the final speed of the boy, follow these steps: 1. Calculate the initial momentum of the dodgeball: \(p_{ball} = (0.5111 \, kg)(23.63 \, m/s)\). 2. Calculate the initial momentum of the boy: \(p_{boy} = (48.95 \, kg)(0 \, m/s) = 0 \, kg \cdot m/s\). 3. Determine the total initial momentum: \(p_{initial} = p_{ball} + p_{boy}\). 4. Calculate the final momentum: \(p_{final} = p_{initial}\). 5. Calculate the final velocity of the boy and dodgeball: \(v_{final} = \frac{p_{final}}{m_{total}}\), where \(m_{total} = m_{ball} + m_{boy}\). 6. Use the calculated values to find the final speed of the boy after catching the dodgeball.

Step-by-step solution

Calculate the initial momentum of the dodgeball

First, we need to find the initial momentum of the dodgeball before it is caught by the boy. The formula for momentum is given by: \(p = mv\), where \(p\) is the momentum, \(m\) is the mass, and \(v\) is the velocity. Convert the mass of the dodgeball to kg: \(m_{ball} = 511.1 \, g = 0.5111 \, kg\). Then, calculate the initial momentum of the dodgeball: \(p_{ball} = (0.5111 \, kg)(23.63 \, m/s)\).

Calculate the initial momentum of the boy

Since the boy is initially at rest, his initial velocity is \(0 \, m/s\). Therefore, his initial momentum is \(p_{boy} = (48.95 \, kg)(0 \, m/s) = 0 \, kg \cdot m/s\).

Determine the total initial momentum

Add the initial momentum of the dodgeball and the initial momentum of the boy to find the total initial momentum: \(p_{initial} = p_{ball} + p_{boy}\).

Calculate the final momentum

According to the conservation of momentum, the final momentum must be equal to the initial momentum: \(p_{final} = p_{initial}\).

Calculate the final velocity of the boy and dodgeball

Now that we know the final momentum, we can calculate the final velocity of the combined system of boy and dodgeball after the catch. We will denote the final velocity as \(v_{final}\). The total mass of the combined system is the sum of the boy's mass and the dodgeball's mass: \(m_{total} = m_{ball} + m_{boy}\). The final velocity can be found by dividing the final momentum by the total mass: \(v_{final} = \frac{p_{final}}{m_{total}}\).

Find the final speed of the boy

Use the calculated values from the previous steps to find the final speed of the boy after catching the dodgeball. This will give us the answer to the problem.