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 32

Chapter 8, Problem 32

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?

Verified step by step guidance

Identify the principle to use: This is a collision problem on a frictionless surface, so the law of conservation of momentum applies. The total momentum before the collision equals the total momentum after the collision.

Write the equation for the total momentum before the collision: \( p_{\text{initial}} = m_1 v_1 + m_2 v_2 \), where \( m_1 = 70.0 \ \text{kg} \), \( v_1 = 4.00 \ \text{m/s} \), \( m_2 = 65.0 \ \text{kg} \), and \( v_2 = -2.50 \ \text{m/s} \) (negative because the second skater is moving to the left).

Write the equation for the total momentum after the collision: \( p_{\text{final}} = (m_1 + m_2) v_f \), where \( v_f \) is the final velocity of the combined skaters (since they grab onto each other, they move together as one mass).

Set the total initial momentum equal to the total final momentum: \( m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \). Solve for \( v_f \) by isolating it: \( v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2} \).

Substitute the given values into the equation: \( v_f = \frac{(70.0)(4.00) + (65.0)(-2.50)}{70.0 + 65.0} \). Simplify the numerator and denominator to find the final velocity \( v_f \), which gives both the magnitude and direction of the skaters' velocity after the collision.

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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 with no external forces, the total momentum before an event must equal the total momentum after the event. In this scenario, the momentum of each skater before the collision must be calculated and summed to find the total momentum, which will then be used to determine their combined velocity after they collide.

Momentum Calculation

Momentum is calculated as the product of an object's mass and its velocity (p = mv). For the two skaters, their individual momenta must be calculated using their respective masses and velocities. The direction of the momentum is also important, as it indicates whether the momentum is positive or negative based on the chosen coordinate system.

Elastic vs. Inelastic Collisions

Inelastic collisions are those in which the colliding objects stick together after the collision, as is the case with the skaters. This type of collision conserves momentum but not kinetic energy. Understanding this distinction is crucial for applying the conservation of momentum correctly to find the final velocity of the combined mass of the skaters after they collide.

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Textbook Question

When cars are equipped with flexible bumpers, they will bounce off each other during low-speed collisions, thus causing less damage. In one such accident, a 1750-kg car traveling to the right at 1.50 m/s collides with a 1450-kg car going to the left at 1.10 m/s. Measurements show that the heavier car's speed just after the collision was 0.250 m/s in its original direction. Ignore any road friction during the collision. Calculate the change in the combined kinetic energy of the two-car system during this collision.

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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 are the magnitude and direction of Daniel's velocity after the collision?

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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?

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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?

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

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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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