Problem 69 - College Physics

Question

\(\bullet\) Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is \(800 \mathrm{N},\) Jane's weight is \(600 \mathrm{N},\) and that of the sleigh is 1000 \(\mathrm{N} .\) They see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity (relative to the ice) of 5.00 \(\mathrm{m} / \mathrm{s}\) at \(30.0^{\circ}\) above the horizontal, and Jane jumps to the right at 7.00 \(\mathrm{m} / \mathrm{s}\) at \(36.9^{\circ}\) above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

Step-by-Step Solution

Answer

0.098 m/s to the right.

Step 1: Understand the Problem

We need to calculate the sleigh's horizontal velocity after Jonathan and Jane jump off. Since the problem occurs on a frictionless surface, we apply the conservation of momentum in the horizontal direction.

Step 2: Calculate Total Mass

Convert the weights to masses by dividing by gravity \( m_{Jonathan} = \frac{800}{9.8} \, \text{kg} = 81.63 \, \text{kg} \), \( m_{Jane} = \frac{600}{9.8} \, \text{kg} = 61.22 \, \text{kg} \), and \( m_{sleigh} = \frac{1000}{9.8} \, \text{kg} = 102.04 \, \text{kg} \).

Step 3: Calculate Initial Momentum

Initially, the total system (Jonathan, Jane, and the sleigh) was at rest, so the total momentum was zero.

Step 4: Calculate Jonathan's Momentum Components

Jonathan's horizontal momentum is \(m_{Jonathan} \times 5.00 \times \cos(30.0^{\circ}) = 81.63 \times 5.00 \times 0.866 = 353.18 \, \text{kg}\textrm{m/s}\). This is directed to the left, thus is negative: \(-353.18 \text{ kg}\cdot \text{m/s}\).

Step 5: Calculate Jane's Momentum Components

Jane's horizontal momentum is \(m_{Jane} \times 7.00 \times \cos(36.9^{\circ}) = 61.22 \times 7.00 \times 0.8 = 343.19 \, \text{kg}\text{m/s}\). This is directed to the right, so it is positive.

Step 6: Apply Conservation of Momentum

By the conservation of momentum, the initial momentum of Jonathan and Jane, plus the momentum of the sleigh, equals zero:\(-353.18 + 343.19 + m_{sleigh} \cdot v_{sleigh} = 0\). Solve for \(v_{sleigh}\): \(v_{sleigh} = \frac{353.18 - 343.19}{102.04} = \frac{9.99}{102.04} = 0.098 \, \text{m/s}\) to the right.

Key Concepts

Physics Problem Solving

Physics often involves understanding complex scenarios by breaking them down into simpler parts. In this exercise, Jonathan and Jane jumping from the sleigh involves momentum conservation concepts on a frictionless surface. By identifying the essential components like velocity, angle, and direction of both people, we can calculate the sleigh's resulting motion.
Start by assessing what you know about the system before and after the event. Initially, the sleigh and its occupants are at rest. After they jump off, the system must conserve momentum. This means any change in their motion must result in the sleigh moving in a way that balances the total momentum to zero.
For effective physics problem-solving:

• Break down the problem into clear steps.
• Identify known and unknown variables.
• Apply relevant physics principles, like the conservation of momentum.

Once you have a clear path, solve using mathematical expressions.

Frictionless Surface

A frictionless surface, like the ice on which the sleigh rests, simplifies calculations by removing forces that could complicate momentum conservation. With no friction, the movement of Jonathan, Jane, and the sleigh are only affected by their interactions and gravity.
This lack of friction means that there is no force opposing the movement caused by Jonathan and Jane jumping, allowing us to directly calculate the sleigh's movement using the momentum principles. Consider friction in general terms as the force hindering motion between surfaces. Here, it's absent, causing no interference and making momentum calculations straightforward.

Horizontal Velocity

Horizontal velocity refers to the component of velocity that runs parallel to the ground, and it is crucial when analyzing motion on surfaces. In this problem, both Jonathan's and Jane's jumps have components of velocity affecting this direction.
Understanding how to find these components involves using trigonometry. We use the given angles to calculate each person's horizontal velocity using cosine functions:
- Jonathan's horizontal velocity is calculated as: \[ 5.00 \times \cos(30.0^{\circ}) \]
- Jane's horizontal velocity is determined similarly: \[ 7.00 \times \cos(36.9^{\circ}) \]
With these, we determine each jump's contribution to the sleigh's horizontal velocity. The sleigh's velocity is then found by ensuring the total horizontal momentum remains zero, accounting for direction.

Weight to Mass Conversion

Weight and mass are distinct, though often confused, concepts in physics. Weight is the gravitational force on an object, typically measured in newtons (N), while mass is the amount of matter in an object, measured in kilograms (kg). Understanding the difference is crucial for correctly applying physics formulas.
To convert weight to mass, use the gravitational acceleration, \( g = 9.8 \, \text{m/s}^2 \). The formula is:
\[ m = \frac{W}{g} \]
where \( m \) is mass and \( W \) is weight.
For this exercise, we use it to find:

• Jonathan's mass: \( \frac{800}{9.8} \approx 81.63 \, \text{kg} \)
• Jane's mass: \( \frac{600}{9.8} \approx 61.22 \, \text{kg} \)
• Sleigh's mass: \( \frac{1000}{9.8} \approx 102.04 \, \text{kg} \)

These values are foundational for calculating momentum.