Question

Jack (mass 55.0 kg) is sliding due east with speed 8.00 m/s on the surface of a frozen pond. He collides with Jill (mass 48.0 kg), who is initially at rest. After the collision, Jack is traveling at 5.00 m/s in a direction 34.0\(^\circ\) north of east. What is Jill's velocity (magnitude and direction) after the collision? Ignore friction.

Short answer

Jill's velocity is approximately 2.57 m/s in a direction 19.3° south of east.

Step-by-step solution

Identify the Conservation Principle

This problem involves a collision, so we need to use the principle of conservation of momentum. The overall momentum before the collision should equal the overall momentum after the collision.

Calculate Initial Momentum

Before the collision, only Jack is moving, so the initial momentum is solely due to Jack. The initial momentum vector can be calculated as follows:\[ p_{initial, Jack} = m_{Jack} \times v_{initial, Jack} = 55.0 \, \text{kg} \times 8.00 \, \text{m/s} = 440 \, \text{kg} \, \text{m/s} \]

Break Down Final Velocities into Components

After the collision, we need to find the components of Jack and Jill's velocities. For Jack:- Eastward (x-direction): \[ v_{Jack, x} = 5.00 \, \text{m/s} \times \cos(34.0^{\circ}) \]- Northward (y-direction): \[ v_{Jack, y} = 5.00 \, \text{m/s} \times \sin(34.0^{\circ}) \]

Write Momentum Equations

Since momentum is conserved separately in each direction, write the equations for x and y components:- In x-direction: \[ m_{Jack} \times v_{Jack, x} + m_{Jill} \times v_{Jill, x} = 440 \, \text{kg} \, \text{m/s} \]- In y-direction:\[ m_{Jack} \times v_{Jack, y} + m_{Jill} \times v_{Jill, y} = 0 \] (because Jill starts from rest and no initial y-component momentum)

Solve for Jill's Velocity Components

From the equations in Step 4, we solve for Jill's velocity components:1. In x-direction:\[ 55.0 \, \text{kg} \times 5.00 \times \cos(34.0^{\circ}) + 48.0 \, \text{kg} \times v_{Jill, x} = 440 \]Solving for \( v_{Jill, x} \) gives:\[ v_{Jill, x} = \frac{440 - 55.0 \times 5.00 \times \cos(34.0^{\circ})}{48.0} \]2. In y-direction:\[ 55.0 \, \text{kg} \times 5.00 \times \sin(34.0^{\circ}) + 48.0 \, \text{kg} \times v_{Jill, y} = 0 \]Solving for \( v_{Jill, y} \) gives:\[ v_{Jill, y} = -\frac{55.0 \times 5.00 \times \sin(34.0^{\circ})}{48.0} \]

Calculate Jill's Resultant Velocity and Direction

Calculate the magnitude of Jill's velocity:\[ v_{Jill} = \sqrt{v_{Jill, x}^2 + v_{Jill, y}^2} \]To find the direction of Jill's velocity, use the arctangent:\[ \theta = \tan^{-1}\left(\frac{v_{Jill, y}}{v_{Jill, x}}\right) \]This angle \( \theta \) describes Jill's direction south of east.

Key concepts

Collision Analysis

When studying physics, understanding collisions is crucial for grasping how forces and momentum interact. A collision occurs when two objects come together with a certain force, causing energy and momentum to be exchanged.
In situations like Jack and Jill's sliding encounter on ice, we're looking at an elastic collision, which assumes no loss of kinetic energy during the event. Here, we apply the conservation of momentum principle because even though forces are exerted, the sum of the initial momenta equals the sum of the final momenta.
This principle allows us to analyze each participant's mass and velocity before and after the collision. With Jack initially in motion, only he contributes to the pre-collision momentum. Post-collision, both Jack and Jill have velocities and contribute to the collective momentum.

Velocity Components

Understanding velocity components is key when analyzing the movement of objects in different directions, especially after a collision.
Velocity can be broken down into at least two perpendicular components, typically horizontal (x-direction) and vertical (y-direction).
In our scenario, after the collision, Jack moves at an angle of 34 degrees north of east, prompting us to calculate both eastward (x) and northward (y) velocity components.

• Eastward component: This reflects how fast Jack moves along the east. It's calculated using the cosine of the given angle and the total velocity.
• Northward component: This indicates Jack's speed going northward. It's found using the sine of the angle and the speed.

These components allow us to apply these principles to examine post-impact velocities, keeping the direction distribution clear for easier calculations.

Momentum Conservation in Physics

The conservation of momentum is a foundational concept in physics that holds that the total momentum of two interacting objects remains constant if no external forces are acting upon them.
For solving a problem like Jack and Jill's collision, we distinguish momentum separately along each axis to account for their individual contributions to the system.

• In the horizontal (x-axis), we apply the initial momentum only from Jack since Jill is at rest initially.
• In the vertical (y-axis), even though Jack has a vertical component, Jill starts with no vertical movement, making her initial vertical momentum zero.

This principle's strength is that it allows us to create a system of equations that makes solving for unknown post-collision velocities possible, maintaining the system's total momentum balance.

Resultant Velocity Calculation

After determining the velocity components for an object, calculating the resultant velocity involves re-combining these components into a single directional speed with a known magnitude and direction.
For Jill, after the collision, we have found her x and y components. To find her total velocity, we utilize the Pythagorean theorem:

• Calculate the magnitude, which is the square root of the sum of the square of Jill's velocity components.

Once the magnitude is known, we can determine the direction by finding the angle using trigonometric functions:

• Use the arctangent function \(\theta = \tan^{-1}\left(\frac{v_{Jill,y}}{v_{Jill,x}}\right)\) to calculate the angle south of east.

This gives a complete view of Jill's speed and how to direct that movement, enabling us to fully understand the dynamics of the collision aftermath.