Problem 97 - University Physics with Modern Physics

Question

Jack and Jill are standing on a crate at rest on the frictionless, horizontal surface of a frozen pond. Jack has mass 75.0 kg, Jill has mass \(45.0 \mathrm{kg},\) and the crate has mass 15.0 \(\mathrm{kg}\) . They remember that they must fetch a pail of water, so each jumps horizontally from the top of the crate. Just after each jumps, that person is moving away from the crate. with a speed of 4.00 \(\mathrm{m} / \mathrm{s}\) relative to the crate. (a) What is the final speed of the crate if both Jack and Jill jump simultaneously and in the same direction? (Hint: Use an inertial coordinate system attached to the ground.) (b) What is the final speed of the crate if Jack jumps first and then a few seconds later Jill jumps in the same direction? (c) What is the final speed of the crate if Jill jumps first and then Jack, again in the same direction?

Step-by-Step Solution

Answer

(a) The crate moves at 32.0 m/s if they both jump simultaneously in the same direction.

Step 1: Analyze Initial Conditions

We have a system consisting of Jack (75.0 kg), Jill (45.0 kg), and the crate (15.0 kg) on a frictionless horizontal surface. Initially, the total momentum of the system is zero as they are at rest.

Step 2: Simultaneous Jump (Part a)

When Jack and Jill jump simultaneously, they push the crate in the opposite direction with a combined relative velocity of 4.00 m/s according to the crate. Using conservation of momentum, the total final momentum of the system must be zero:\[(75.0 \times 4.00) + (45.0 \times 4.00) = 15.0 \times v_{crate}\]This simplifies to: \[v_{crate} = \frac{(75.0 + 45.0) \times 4.00}{15.0}\]Calculating gives \[v_{crate} = 32.0\]

Key Concepts

Inertial Coordinate System

Understanding an inertial coordinate system is fundamental when dealing with problems in physics related to motion and forces. These systems serve as reference frames where Newton's laws of motion hold true. This can be a stationary frame of reference, like the ground in the Jack and Jill problem, or one that moves at a constant velocity.

Inertial coordinate systems do not accelerate, ensuring that objects within them follow predictable paths as described by the principles of physics. In the original problem, you are advised to use an inertial coordinate system attached to the ground, which remains stationary. This means you view the motion from a fixed point without the influence of external accelerations.

• The initial total momentum, when Jack, Jill, and the crate are still, is zero, reflecting the rest condition in the inertial system.
• As they jump off the crate, each acts as an applied force that changes the state of inertia in accordance with conservation principles.

Using these principles makes it easier to calculate the changes in the system's dynamic, such as the resultant speed of the crate.

Frictionless Surface

A frictionless surface simplifies complex physics problems by eliminating the resistive forces that would otherwise complicate calculations. In the Jack and Jill scenario, the frozen pond acts as a frictionless surface, which means no force opposes the movement caused by their jumps.

On a frictionless surface, all kinetic actions result in opposite reactions without any energy lost to frictional heat or resistance.

• This absence of friction ensures that when Jack and Jill jump from the crate, the resulting horizontal motion of the crate is unaffected by external frictional forces.
• Calculating momentum shifts becomes simpler as the potential energy loss through friction is nonexistent.

This perfect frictionless environment allows us to use conservation of momentum accurately, predicting how the actions of the jumping individuals translate into motion, such as the velocity changes in the crate.

Relative Velocity

Relative velocity is a concept that helps understand the motion of an object as observed from another moving object. In problems like this one, where individuals are moving in coordination with objects like the crate, understanding relative velocity is key to solving the dynamics of motion.

In the given exercise, the jumps are described as having a velocity relative to the crate. This means:

• The velocity (4.00 m/s) is measured in terms of how fast Jack and Jill are moving away from the crate, not in relation to the ground.
• You must consider both the motion of Jack and Jill and the resulting motion of the crate within the same inertial frame.

Relative velocity helps you translate these personal movements into the ground's inertial frame, which is crucial for calculating the final states of motion within the problem. This ensures a consistent application of conservation laws throughout the problem.