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 27a

Chapter 8, Problem 27a

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?

Verified step by step guidance

Step 1: Identify the type of collision. Since the problem involves two skaters colliding and moving afterward, this is a two-dimensional collision. Momentum conservation will be applied separately in the x-direction and y-direction.

Step 2: Write the conservation of momentum equations. Momentum is conserved in both the x and y directions. Let Daniel's velocity after the collision be \( v_D \) and Rebecca's velocity after the collision be \( v_R \). Use the following equations: \( m_R v_{R, initial} = m_R v_{R, x} + m_D v_{D, x} \) for the x-direction, and \( 0 = m_R v_{R, y} + m_D v_{D, y} \) for the y-direction. Here, \( v_{R, x} \) and \( v_{R, y} \) are Rebecca's velocity components after the collision, and \( v_{D, x} \) and \( v_{D, y} \) are Daniel's velocity components after the collision.

Step 3: Resolve Rebecca's velocity into components. Use trigonometry to find \( v_{R, x} \) and \( v_{R, y} \): \( v_{R, x} = v_R \cos(\theta) \) and \( v_{R, y} = v_R \sin(\theta) \), where \( v_R = 8.00 \, \text{m/s} \) and \( \theta = 53.1^\circ \).

Step 4: Solve for Daniel's velocity components. Substitute the known values into the momentum conservation equations to find \( v_{D, x} \) and \( v_{D, y} \). Then, calculate the magnitude of Daniel's velocity using \( v_D = \sqrt{v_{D, x}^2 + v_{D, y}^2} \), and find the direction using \( \tan^{-1}(v_{D, y} / v_{D, x}) \).

Step 5: Calculate the change in kinetic energy. The initial kinetic energy is \( KE_{initial} = \frac{1}{2} m_R v_{R, initial}^2 \). The final kinetic energy is \( KE_{final} = \frac{1}{2} m_R v_R^2 + \frac{1}{2} m_D v_D^2 \). The change in kinetic energy is \( \Delta KE = KE_{final} - KE_{initial} \).

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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 must equal the total momentum after the event, provided no external forces act on it. In this scenario, the momentum of Daniel and Rebecca before and after the collision must be calculated to determine Daniel's velocity after the collision.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 1/2 mv², where m is mass and v is velocity. Understanding how to calculate the kinetic energy of both skaters before and after the collision is essential for determining the change in total kinetic energy resulting from the collision.

Vector Components

In physics, vectors have both magnitude and direction, and can be broken down into components along the axes of a coordinate system. After the collision, Rebecca's velocity is given at an angle, requiring the use of trigonometric functions to resolve her velocity into horizontal and vertical components, which is crucial for applying conservation of momentum in both dimensions.

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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

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. What was the speed of the lighter car just after the collision?

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Textbook Question

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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?

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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.

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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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