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 22a

Chapter 8, Problem 22a

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?

Verified step by step guidance

Step 1: Identify the type of collision. Since the problem involves cars bouncing off each other, it is an elastic collision. In such cases, momentum is conserved, and we can use the principle of conservation of momentum to solve for the unknowns.

Step 2: Write the equation for the conservation of momentum. The total momentum before the collision equals the total momentum after the collision. Mathematically, this is expressed as:

m

₁v_1i + m₂v_2i = m₁v_1f + m₂v_2f, where

m

₁ and

m

₂ are the masses of the two cars,

v

_1i and

v

_2i are their initial velocities, and

v

_1f and

v

_2f are their final velocities.

Step 3: Substitute the known values into the momentum conservation equation. Use

m

₁ = 1750 kg,

v

_1i = 1.50 m/s,

m

₂ = 1450 kg,

v

_2i = -1.10 m/s (negative because it is to the left), and

v

_1f = 0.250 m/s. Solve for

v

_2f, the final velocity of the lighter car.

Step 4: Calculate the change in the combined kinetic energy of the system. The kinetic energy before the collision is given by

K_{i} = 0.5 m

₁v_1i² + 0.5m₂v_2i², and the kinetic energy after the collision is

K_{f} = 0.5 m

₁v_1f² + 0.5m₂v_2f². The change in kinetic energy is then

ΔK = K_{f} - K_{i}

Step 5: Substitute the known values into the kinetic energy equations and solve for the change in kinetic energy. Use the previously calculated value of

v

_2f to complete the calculation.

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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 a collision is equal to the total momentum after the collision. This is crucial for analyzing collisions, as it allows us to set up equations based on the masses and velocities of the colliding objects to find unknown quantities, such as the final velocity of the lighter car in this scenario.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated using the formula KE = 0.5 * m * v^2, where m is mass and v is velocity. In collision problems, it's important to compare the total kinetic energy before and after the collision to determine how much energy is conserved or lost, which can indicate the nature of the collision (elastic or inelastic).

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic based on whether kinetic energy is conserved. In elastic collisions, both momentum and kinetic energy are conserved, while in inelastic collisions, momentum is conserved but kinetic energy is not. Understanding this distinction helps in analyzing the outcomes of collisions, such as the one described, where the cars bounce off each other, suggesting an inelastic interaction.

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Intro To Elastic Collisions

Related Practice

Textbook Question

On a frictionless, horizontal air table, puck A (with mass 0.250 kg) is moving toward puck B (with mass 0.350 kg), which is initially at rest. After the collision, puck A has a velocity of 0.120 m/s to the left, and puck B has a velocity of 0.650 m/s to the right. What was the speed of puck A before the 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. 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

You are standing on a sheet of ice that covers the football stadium parking lot in Buffalo; there is negligible friction between your feet and the ice. A friend throws you a 0.600-kg ball that is traveling horizontally at 10.0 m/s. Your mass is 70.0 kg. If the ball hits you and bounces off your chest, so afterward it is moving horizontally at 8.0 m/s in the opposite direction, what is your speed after the collision?

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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