Ch 08: Momentum, Impulse, and Collisions

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 08: Momentum, Impulse, and Collisions

Problem 39

Chapter 8, Problem 39

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° north of east. What is Jill's velocity (magnitude and direction) after the collision? Ignore friction.

Verified step by step guidance

Step 1: Identify the type of collision. Since the problem involves two objects colliding and no external forces are acting (ignoring friction), this is a conservation of momentum problem. Momentum is conserved in both the x-direction (east-west) and the y-direction (north-south).

Step 2: Write the equation for conservation of momentum in the x-direction. The total momentum before the collision in the x-direction is the momentum of Jack (mass × velocity) since Jill is initially at rest. After the collision, the x-component of Jack's momentum and Jill's momentum must sum to the initial momentum. Use the equation: \( m_{Jack} v_{Jack, initial} = m_{Jack} v_{Jack, final} \cos(\theta) + m_{Jill} v_{Jill} \cos(\phi) \), where \( \theta \) is Jack's angle (34.0° north of east) and \( \phi \) is Jill's unknown direction.

Step 3: Write the equation for conservation of momentum in the y-direction. Since there is no initial momentum in the y-direction (north-south), the total momentum in the y-direction after the collision must be zero. Use the equation: \( m_{Jack} v_{Jack, final} \sin(\theta) = m_{Jill} v_{Jill} \sin(\phi) \).

Step 4: Solve the system of equations. You now have two equations: one for the x-direction and one for the y-direction. Solve these equations simultaneously to find Jill's velocity \( v_{Jill} \) and direction \( \phi \). Use trigonometric relationships to isolate \( v_{Jill} \) and \( \phi \).

Step 5: Interpret the results. Once you solve for \( v_{Jill} \) (magnitude of Jill's velocity) and \( \phi \) (direction of Jill's velocity), ensure the direction is expressed relative to east (e.g., degrees north or south of east). This will give you the final answer in terms of magnitude and direction.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Momentum

The principle of conservation of momentum states that in a closed system, the total momentum before an event (like a collision) is equal to the total momentum after the event. This is crucial for solving collision problems, as it allows us to set up equations based on the initial and final velocities and masses of the objects involved.

Vector Components

In physics, vectors can be broken down into their components along the axes of a coordinate system. For this problem, we can resolve the velocities into east-west (x-axis) and north-south (y-axis) components. This simplification is essential for accurately calculating the resultant velocity of Jill after the collision.

Collision Types

Collisions can be classified as elastic or inelastic, with inelastic collisions conserving momentum but not kinetic energy. In this scenario, since Jack and Jill collide and move together afterward, we assume an inelastic collision, which affects how we calculate the final velocities and directions of the objects involved.

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Related Practice

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Textbook Question

Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1° from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. What is the change in total kinetic energy of the two skaters as a result of the collision?

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