Ch 08: Momentum, Impulse, and Collisions

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th 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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

16m

Was this helpful?

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.

Recommended video:

Guided course

04:47

Overview of Collision Types

Related Practice

Textbook Question

To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600-g falcon flying at 20.0 m/s hit a 1.50-kg raven flying at 9.0 m/s. The falcon hit the raven at right angles to its original path and bounced back at 5.0 m/s. (These figures were estimated by the author as he watched this attack occur in northern New Mexico.) By what angle did the falcon change the raven's direction of motion?

views

Textbook Question

To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a 600-g falcon flying at 20.0 m/s hit a 1.50-kg raven flying at 9.0 m/s. The falcon hit the raven at right angles to its original path and bounced back at 5.0 m/s. (These figures were estimated by the author as he watched this attack occur in northern New Mexico.) What was the raven's speed right after the collision?

views

Textbook Question

A small rocket burns 0.0500 kg of fuel per second, ejecting it as a gas with a velocity relative to the rocket of magnitude 1600 m/s. What is the thrust of the rocket?

views

Textbook Question

Two skaters collide and grab on to each other on frictionless ice. One of them, of mass 70.0 kg, is moving to the right at 4.00 m/s, while the other, of mass 65.0 kg, is moving to the left at 2.50 m/s. What are the magnitude and direction of the velocity of these skaters just after they collide?

views

Textbook Question

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/s. They hold fast to each other after they collide. How much mechanical energy dissipates during this play?

views

Textbook Question

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/s. They hold fast to each other after they collide. Find the magnitude and direction of the velocity of these free-spirited otters right after they collide.

views