Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

문제 66

11장, 문제 66

Consider a partially elastic collision in which ball A of mass m with initial velocity (v_ix)_A collides with stationary ball B, also of mass m, and in which 1/4 of the mechanical energy is dissipated as thermal energy. Find expressions for the final velocities of each ball. Hint: Mathematically there are two solutions; however, one of them is physically impossible.

검증된 단계별 안내

Step 1: Begin by applying the principle of conservation of momentum. Since the collision occurs in one dimension, the total momentum before and after the collision must be equal. Write the equation:

m_{A} v_{ix} A = m_{A} v_{fx} A + m_{B} v_{fx} B

, where

v_{fx}

represents the final velocities of balls A and B.

Step 2: Use the principle of conservation of energy, but account for the fact that 1/4 of the mechanical energy is dissipated as thermal energy. Write the equation for the initial kinetic energy and the remaining kinetic energy after the collision:

(3 / 4)

of the initial kinetic energy is conserved. Express this as:

(3 / 4) m_{A} v_{ix} A^2 = m_{A} v_{fx} A^2 + m_{B} v_{fx} B^2

Step 3: Solve the system of equations formed by the conservation of momentum and the modified conservation of energy. Substitute the expression for

v_{fx} B

from the momentum equation into the energy equation to eliminate one variable. This will allow you to solve for

v_{fx} A

Step 4: After solving for

v_{fx} A

, substitute this value back into the momentum equation to find

v_{fx} B

. Ensure that both solutions satisfy the physical constraints of the problem, such as the direction of motion and the dissipation of energy.

Step 5: Identify the physically impossible solution by checking whether it violates any physical principles, such as conservation laws or the dissipation of energy. Discard the invalid solution and retain the valid one as the final expressions for the velocities of balls A and B.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

16m

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Conservation of Momentum

In a collision, the total momentum of a closed system remains constant if no external forces act on it. This principle allows us to relate the initial and final velocities of colliding objects. For two colliding balls, the equation m(vix)A + m(0) = m(vfx)A + m(vfx)B can be used, where vfx represents the final velocities of balls A and B.

Kinetic Energy and Elasticity

Kinetic energy is the energy of motion, given by the formula KE = 1/2 mv^2. In elastic collisions, kinetic energy is conserved, while in partially elastic collisions, some energy is transformed into other forms, such as thermal energy. In this scenario, 1/4 of the mechanical energy is lost, which affects the final velocities of the balls.

Physical Feasibility of Solutions

When solving for final velocities in collision problems, it is essential to evaluate the physical feasibility of the solutions. One solution may yield negative velocities or violate conservation laws, indicating it is not physically possible. Thus, only the solution that adheres to the principles of physics and the context of the problem should be accepted.

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