Question

A boy is playing rollerblade dodgeball. His rollerblades are frictionless, and he is initially at rest. A dodgeball with a mass of \(515.1 \mathrm{~g}\) is thrown directly at him with a speed of \(24.91 \mathrm{~m} / \mathrm{s}\). He catches the dodgeball and then moves with a speed of \(0.2188 \mathrm{~m} / \mathrm{s}\). What is the mass of the boy?

Short answer

Answer: The mass of the boy is approximately 58.62 kg.

Step-by-step solution

Write down the conservation of momentum equation

The conservation of momentum equation states that the total momentum before the collision is equal to the total momentum after the collision. Mathematically, it can be written as: \(m_a v_a + m_b v_b = (m_a + m_b) v'\) Here, \(m_a\) = mass of the dodgeball, \(v_a\) = speed of the dodgeball, \(m_b\) = mass of the boy (which we need to find), \(v_b\) = speed of the boy (initially at rest, so \(v_b=0\)), \(v'\) = combined speed of the boy and dodgeball after collision.

Substitute the given values into the equation

Now we substitute the given values into the conservation of momentum equation: \((0.5151 \mathrm{~kg}) (24.91 \mathrm{~m/s}) + m_b (0 \mathrm{~m/s}) = (0.5151 \mathrm{~kg} + m_b) (0.2188 \mathrm{~m/s})\)

Solve for the mass of the boy, \(m_b\)

Simplifying and solving for \(m_b\), we get: \(0.5151 * 24.91 = 0.5151 * 0.2188 + m_b * 0.2188\) \(0.5151 * 24.91 - 0.5151 * 0.2188 = m_b * 0.2188\) Now, divide both sides by \(0.2188\) to isolate \(m_b\): \(m_b = \frac{0.5151 * 24.91 - 0.5151 * 0.2188}{0.2188}\) Calculate the value of \(m_b\): \(m_b = \frac{12.8261}{0.2188} = 58.62 \thinspace \mathrm{kg}\) (rounded to 2 decimal places)

State the answer

The mass of the boy is approximately \(58.62 \thinspace \mathrm{kg}\).

Key concepts

Momentum in Collisions

When we talk about momentum in collisions, we're looking at how objects interact with each other when they come into contact. In physics, momentum is defined as the product of the mass of an object and its velocity, often represented by the formula \( p = mv \).

In the event of a collision, such as a dodgeball being caught by a person on rollerblades, the principle we are most concerned with is whether momentum is conserved or not. In a closed system, where no external forces act upon the objects involved, the total momentum before the collision must equal the total momentum after the collision. It's a bit like having a set number of building blocks; you can rearrange them however you like, but you can't change the total amount you have unless you add or remove some from your set.

In our rollerblade dodgeball scenario, the amazing part is that by catching the ball, the boy's motion afterward - his speed and direction - tells us a story about his mass. It's a practical example that enables us to visualize momentum conservation and use it to find out something unknown (the boy’s mass) given the other known values.

Physics Problem Solving

When approaching physics problem solving, it's crucial to have a clear strategy. First, identify what is asked and what information is provided. Then, determine the relevant physics principles that can be applied to the situation. In our case, we are concerned with a momentum-related problem, so we identify the conservation of momentum as the guiding principle.

Breaking down the problem into steps helps to manage it more effectively. For instance, writing out the appropriate equation, as seen in our dodgeball scenario, and substituting the given values systematically can help clarify the process.

Drawing a Diagram

Often, sketching a simple diagram can be incredibly helpful for visual learners. It's a common and useful practice in physics to illustrate the before and after states of a collision or interaction.

Checking Units

Always ensure the units of the given quantities are compatible and, if necessary, perform unit conversions before plugging the values into equations.

Focusing on Concepts

Rather than getting lost in the math, understanding why each step is taken from a conceptual standpoint can reinforce the learning experience and can help in retaining the fundamental physics principles.

Momentum Conservation Equation

The momentum conservation equation is a beautiful demonstration of Newton’s third law, which tells us that for every action, there's an equal and opposite reaction. Understanding this equation is not just about knowing the mathematical formula but also about comprehending its physical basis.

Let's explore the equation used in our exercise: \[ m_a v_a + m_b v_b = (m_a + m_b) v' \] This describes a situation where two objects, in this case, a dodgeball (\(m_a\)) and a boy (\(m_b\)), interact. Because the boy is initially at rest, his starting velocity (\(v_b\)) is zero. After he catches the ball, they move together with a new velocity (\(v'\)). This illustrates the law of conservation of momentum in a closed system.

Solving this equation can seem daunting, but breaking it down step by step simplifies it. It's all about isolating the variable we're trying to find – in this case, the boy's mass. By rearranging the equation and solving it methodically, just like in a recipe, we reach the answer. Understanding that this process applies universally, not just to dodgeballs and rollerblades, but to car crashes, rocket launches, or even celestial events, opens a world of physics applications.