Ch 11: Impulse and Momentum

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 11: Impulse and Momentum

Problem 64a

Chapter 11, Problem 64a

A 100 g ball moving to the right at 4.0 m/s collides head-on with a 200g ball that is moving to the left at 3.0 m/s. If the collision is perfectly elastic, what are the speed and direction of each ball after the collision?

Verified step by step guidance

Step 1: Begin by understanding the concept of a perfectly elastic collision. In such collisions, both momentum and kinetic energy are conserved. Use these two conservation laws to solve the problem.

Step 2: Write the equation for conservation of momentum. The total momentum before the collision equals the total momentum after the collision. Use the formula:

m_{1} v_{1} + m_{2} v_{2} = m_{1} v_{1}' + m_{2} v_{2}'

, where

v_{1}

and

v_{2}

are the initial velocities, and

v_{1}'

and

v_{2}'

are the final velocities.

Step 3: Write the equation for conservation of kinetic energy. The total kinetic energy before the collision equals the total kinetic energy after the collision. Use the formula:

(1 / 2) m_{1} v_{1}^2 + (1 / 2) m_{2} v_{2}^2 = (1 / 2) m_{1} v_{1}'^2 + (1 / 2) m_{2} v_{2}'^2

Step 4: Substitute the given values into the equations. The masses are

m_{1} = 0.1

kg and

m_{2} = 0.2

kg, and the initial velocities are

v_{1} = 4.0

m/s and

v_{2} = - 3.0

m/s. Solve the system of equations to find the final velocities

v_{1}'

and

v_{2}'

Step 5: Use algebraic manipulation to solve the system of equations derived from the conservation laws. This will yield the final velocities of both balls after the collision. Ensure the directions are consistent with the signs of the velocities.

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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 objects involved.

Elastic Collisions

In a perfectly elastic collision, both momentum and kinetic energy are conserved. This means that not only do the objects bounce off each other without losing energy, but their total kinetic energy before and after the collision remains constant, which is essential for calculating the final velocities of the colliding objects.

Velocity and Direction

Velocity is a vector quantity that includes both speed and direction. In collision problems, it is important to consider the direction of each object's velocity, as this affects the overall momentum and energy calculations. The final velocities after the collision will depend on the initial velocities and the masses of the colliding objects.

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Textbook Question

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.

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