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 22b

Chapter 8, Problem 22b

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.

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: To calculate the change in kinetic energy, first find the total initial kinetic energy and the total final kinetic energy of the system. The kinetic energy is given by the formula

K = 1/2 m v

². Compute the initial kinetic energy for both cars using their initial velocities, and the final kinetic energy using their final velocities.

Step 5: Subtract the total final kinetic energy from the total initial kinetic energy to find the change in kinetic energy of the system. This value represents the energy lost (or gained) during the collision, which could be due to factors like sound, heat, or deformation of the bumpers.

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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 collisions, this means that the sum of the products of mass and velocity for all objects involved remains constant. This concept is crucial for solving problems involving collisions, as it allows us to relate the velocities of the colliding objects before and after the impact.

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. In collision problems, understanding kinetic energy helps assess how energy is transferred or transformed during the event. It is important to analyze the change in kinetic energy to determine how much energy is lost or gained, which can indicate whether the collision is elastic or inelastic.

Elastic vs. Inelastic Collisions

Collisions can be classified as elastic or inelastic based on how they conserve kinetic energy. In elastic collisions, both momentum and kinetic energy are conserved, while inelastic collisions conserve momentum but not kinetic energy. The distinction is important for analyzing the outcomes of collisions, as it affects how the velocities of the objects change and how much energy is dissipated as heat or deformation during the impact.

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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

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

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?

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 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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