Question

A \(53.53-\mathrm{kg}\) boy is playing rollerblade dodgeball. His rollerblades are frictionless, and he is initially at rest. A dodgeball with a mass of \(513.1 \mathrm{~g}\) is thrown directly at him, and he catches it. After catching the ball, the boy moves with a speed of \(0.2304 \mathrm{~m} / \mathrm{s}\). With what speed was the dodgeball thrown?

Short answer

Based on the given information and the conservation of momentum law, the initial velocity of the dodgeball when it was thrown is approximately 24.22 m/s.

Step-by-step solution

Convert the dodgeball mass to kg

We have the mass of the dodgeball as 513.1 g. To work with SI units, we need to convert it to kg. We can do this by dividing its mass by 1000: Dodgeball mass: \(\frac{513.1}{1000} = 0.5131 \mathrm{~kg}\)

Write down the conservation of momentum equation

According to the conservation of momentum law, Initial total momentum = Final total momentum \(m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f\) Here, \(m_1\) is the mass of the boy, \(v_1\) is the initial velocity of the boy, \(m_2\) is the mass of the dodgeball, \(v_2\) is the initial velocity of the dodgeball, and \(v_f\) is the final velocity of both after the collision. Since the boy is initially at rest, \(v_1 = 0\). We are given the final velocity \(v_f = 0.2304 \mathrm{~m/s}\) and we found out the mass of dodgeball in step 1. We can rearrange the equation to solve for the initial velocity of the dodgeball, \(v_2\).

Rearrange the equation to solve for the initial velocity of the dodgeball

The conservation of momentum equation can be rearranged to solve for the initial velocity of the dodgeball, \(v_2\): \(v_2 = \frac{(m_1 + m_2) v_f - m_1 v_1}{m_2}\) Now, we plug in the given values and solve for \(v_2\).

Calculate the initial velocity of the dodgeball

We substitute the given values and calculate \(v_2\): \(v_2 = \frac{(53.53\mathrm{~kg} + 0.5131\mathrm{~kg})(0.2304\mathrm{~m/s}) - 53.53\mathrm{~kg}(0)}{0.5131\mathrm{~kg}}\) \(v_2 = \frac{12.4257\mathrm{~kg\cdot m/s}}{0.5131\mathrm{~kg}}\) \(v_2 = 24.222\mathrm{~m/s}\) So, the initial velocity of the dodgeball when it was thrown is approximately \(24.22 \mathrm{~m/s}\).

Key concepts

Momentum Calculation

Momentum is a fundamental concept in physics that combines an object's mass and its velocity to represent its motion. The momentum of an object is calculated by the formula \( p = m \times v \), where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity. In the context of our dodgeball scenario, the principle of conservation of momentum is key. Before and after the boy catches the dodgeball, the total momentum must remain the same, provided no external forces act upon the system.

This principle allows us to set up the equation \( m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \) as seen in Step 2 of the solution. The next step is to input the right values into this equation, taking care to convert any measurements to SI units to ensure consistency. For instance, when we talk about velocity, we want to make sure we're using meters per second (m/s), and for mass, the kilograms (kg) unit is standard. Only by using the correct units can we ensure the accuracy of our momentum calculation.

Initial Velocity

Initial velocity is the speed at which an object is moving when it first comes under consideration. In physics problems, determining this value often involves understanding the conditions at the start of a scenario, before any changes in motion occur. In the dodgeball example, the key requirement is to find the speed at which the dodgeball was thrown, known as its initial velocity \( v_2 \).

In scenarios where the conservation of momentum applies, we can find the initial velocity by rearranging our momentum conservation equation, putting \( v_2 \) on one side. Given that the boy was initially at rest (\( v_1 = 0 \)), it reduces the complexity of our calculation. Following this, we substitute known values for the boy's mass (\( m_1 \)), the combined final velocity (\( v_f \)), and the mass of the dodgeball (\( m_2 \) - after converting it to kilograms) to find the initial velocity. The execution of this step was beautifully illustrated in Step 3 and Step 4 of the solution.

Mass Conversion

Mass conversion is a simple yet important aspect of solving physics problems, as it ensures that calculations are done using the consistent and correct units, typically kilograms in the International System of Units (SI). In the dodgeball problem, we encounter the need for mass conversion at the very start. The mass of the dodgeball is initially given in grams, which is not the SI unit for mass.

As such, a crucial first step was converting the dodgeball's mass from grams to kilograms by dividing by 1000, as seen in Step 1 of the solution (\( \frac{513.1}{1000} = 0.5131 \mathrm{~kg} \)). This conversion ensures compatibility when we apply the conservation of momentum equation. Without such conversions, our calculations could lead to incorrect results, potentially confusing the difference between mass and weight as well. Always remember to convert all mass measurements to kilograms when preparing to perform any momentum-related calculations in physics.