Problem 36 - University Physics with Modern Physics

Question

Stuck in the middle of a frozen pond with only your physics book, you decide to put physics in action and throw the 5.00 -kg book. If your mass is \(62.0 \mathrm{~kg}\) and you throw the book at \(13.0 \mathrm{~m} / \mathrm{s}\), how fast do you then slide across the ice? (Assume the absence of friction.)

Step-by-Step Solution

Answer

Answer: 1.05 m/s

Step 1: Write the equation for conservation of momentum

The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces act on it. In this case, the combined momentum of the person and the book before the throw is equal to the combined momentum after the throw. We can write this as: $$m_p \times v_p + m_b \times v_b = m_p \times v'_p + m_b \times v'_b$$ Where: - \(m_p\) is the person's mass (\(62.0 \mathrm{~kg}\)) - \(v_p\) is the person's initial velocity (before the throw, \(0 \mathrm{~m/s}\)) - \(m_b\) is the book's mass (\(5.00 \mathrm{~kg}\)) - \(v_b\) is the book's initial velocity (before the throw, \(0 \mathrm{~m/s}\)) - \(v'_p\) is the person's final velocity (after the throw) - \(v'_b\) is the book's final velocity (after the throw, \(13.0 \mathrm{~m/s}\))

Step 2: Calculate the initial total momentum of the system

Before the throw, both the person and the book are at rest, so their initial velocities (\(v_p, v_b\)) are both \(0 \mathrm{~ m/s}\). Hence, the initial momentum of the system is: $$m_p \times v_p + m_b \times v_b = 62.0 \mathrm{~kg} \times 0 \mathrm{~m/s} + 5.00 \mathrm{~kg} \times 0 \mathrm{~m/s} = 0$$

Step 3: Solve for the person's final velocity

Since the initial total momentum of the system is \(0\), and we know the final momentum of the book, we can solve for the person's final velocity: $$m_p \times v'_p + m_b \times v'_b = 0 \Rightarrow v'_p = -\frac{m_b \times v'_b}{m_p}$$ Substitute the values: $$v'_p = -\frac{5.00 \mathrm{~kg} \times 13.0 \mathrm{~m/s}}{62.0 \mathrm{~kg}}$$

Step 4: Calculate the person's final velocity

Now, we can perform the calculation to find the person's final velocity: $$v'_p = -\frac{5.00 \times 13.0}{62.0} \approx -1.05 \mathrm{~m/s}$$ The negative sign indicates that the person is moving in the opposite direction of the book's throw. The person's final velocity is approximately \(1.05 \mathrm{~m/s}\) in the opposite direction of the thrown book.

Key Concepts

Momentum

Imagine you are gliding effortlessly on ice - this scenario is the perfect setting to demonstrate a crucial concept in mechanical physics known as momentum. In the context of the given exercise, when you throw your physics book across the ice, you're actually observing the law of conservation of momentum in action.

Momentum, symbolized by the letter 'p', is defined as the mass of an object multiplied by its velocity (\(p = m \times v\)). This concept tells us how much 'oomph' an object has when it's moving. Heavy and fast objects have large amounts of momentum. When you’re at rest in the middle of the ice, your momentum is zero - no mass is moving. The book, too, starts with zero momentum.

However, once you throw the book, both you and the book acquire momentum. The book flies off one way, and you slide across the ice in the opposite direction. Even though the directions are opposite, the product of mass and velocity (momentum) for you and the book remain balanced. This balance showcases that momentum has both magnitude and direction, making it a vector quantity.

Mechanical Physics

Mechanical physics, a branch often referred to as classical mechanics, is the foundation of understanding movements and forces on objects. The frozen pond scenario elegantly illustrates the principles of mechanics. When you throw the book, you’re not just moving randomly - you're demonstrating one of the defining principles of mechanical physics: for every action, there’s an equal and opposite reaction.

In our no-friction ice pond, that reaction is you sliding in the opposite direction to the thrown book. Mechanical physics gives you the tools to predict the speed and direction you'll move based on the mass of the book, the force of your throw, and your own mass. You're conducting a real-world physics experiment without even realizing it, which incidentally, could help pass the time on that hypothetical frozen pond.

Newton's Laws of Motion

Newton's laws of motion form the core of the concepts you're grappling with on this fictional frozen pond. The first law (also known as the law of inertia), tells us that an object at rest will stay at rest unless acted upon by an external force - that’s why you're initially stationary on the ice.

But then you throw the book (demonstrating the third law), and for every action (the force of the throw), there is an equal and opposite reaction (you sliding backwards). You can calculate the effect of this action and reaction using the second law, which connects force, mass, and acceleration (\(F = m \times a\)). However, because the surface is frictionless, once you're in motion, there’s nothing to stop you - so you keep moving at a constant velocity.

These laws lay the groundwork for understanding not just motion on ice, but all motion in the universe. The mathematical beauty of Newton's formulations allows us to predict movements, from the smallest particles to the largest celestial bodies, with remarkable precision.