Question

An \(84.4\) -kg man is standing at the rear of a \(425-\mathrm{kg}\) iceboat that is moving at \(4.16 \mathrm{~m} / \mathrm{s}\) across ice that may be considered to be frictionless. He decides to walk to the front of the \(18.2-\mathrm{m}\) -long boat and does so at a speed of \(2.08 \mathrm{~m} / \mathrm{s}\) with respect to the boat. How far does the boat move across the ice while he is walking?

Short answer

The calculation will give us the exact distance the boat moves while the man is walking, by using the conservation of momentum principle and solving for the boat's velocity change due to man's velocity changed.

Step-by-step solution

The initial and final momentums

Firstly, calculate the initial momentum of the system (man and boat) before the man starts walking: \( P_{initial} = (mass_{man} + mass_{boat}) * velocity_{boat} = (84.4 kg + 425 kg) * 4.16 m/s = 2118.64 kg.m/s \). Then, calculate the final momentum of the system after the man has reached the front of the boat, which will still equal the initial momentum.

The changed velocities

The final momentum of the system will be the sum of momentum of the man and that of the boat: \( mass_{man} * velocity_{man} + mass_{boat} * velocity_{boat(new)} = 2118.64 kg.m/s \). Here, \( velocity_{man} = 2.08 m/s \) is the velocity of man relative to the boat. The \( velocity_{boat(new)} \) is what we need to find in order to calculate how far the boat moves during the period when the man walks. Solve this equation to find \( velocity_{boat(new)} \).

The travelled distance

Calculate how long it takes the man to walk to the front of the boat: \( time = distance / speed = 18.2 m / 2.08 m/s = 8.75 s \). Then, multiply this time by \( velocity_{boat(new)} \) to get the distance the boat moves while the man is walking.

Key concepts

Frictionless Ice

In the world of physics, frictionless surfaces make calculations straightforward by ignoring the effects of friction. Friction is a force that opposes motion. When solving problems on surfaces where friction is absent, like frictionless ice, this complexity is removed. As a result, the motion of objects is only influenced by other factors, such as velocity and mass.
In the context of our problem, the ice is considered frictionless, making it a perfect playground for conservation of momentum. The absence of friction ensures that once the iceboat starts moving, it keeps moving at the same speed unless acted upon by another force. This simplification allows for a clearer analysis of how the man's movement affects the boat's motion.
Do keep in mind, in real-world scenarios, friction always plays some role; however, neglecting it for certain problems helps simplify and focus on the central concepts. Using frictionless models can aid in understanding basic physics principles before taking more complex real-world factors into account.

Relative Velocity

Understanding relative velocity is essential when comparing the speed of one object with respect to another. In this problem, the man's speed is specified as being relative to the boat. This means his velocity is measured from the perspective that the boat itself provides.
Here's a simple explanation: Imagine standing still on your friend's truck as it rolls down a hill. To someone outside, you both move together. But if you start walking inside the truck, your speed relative to the truck is how quickly you move from one end to the other.
This concept allows us to understand not just how fast something moves, but how its motion compares from a different perspective. By mastering this idea, you can effectively analyze more complex scenarios like our iceboat problem, where knowing the man's speed concerning the boat allows for solving relative motion issues in a seemingly effortless way.

Physics Problem Solving

Physics problem-solving often starts with identifying what is known and what needs to be determined. Here, the challenge is to determine how far the boat moves while the man walks. To approach this:

• Understand the given data, like the masses and velocities.
• List known formulas, such as conservation of momentum: \( P_{initial} = P_{final} \).
• Apply these formulas step-by-step, aligning each variable with its real-world counterpart.
• Avoid rushing to a solution; always cross-check calculations to ensure they align with physics concepts.
• Try to express your quantities in terms of what you're trying to find, like distance in relation to time and velocity.

This problem-solving checklist helps transform theoretical knowledge into practical understanding. Through structured analysis, one can unravel intricate issues and gain confidence in tackling a wide range of physics challenges.