Elastic Collisions in One Dimension: Velocity Calculations

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Elastic Collisions in One Dimension: Velocity Calculations

In one-dimensional elastic collisions, the momentum and kinetic energy of the system remain constant before and after the collision. This results in specific relationships between the initial and final velocities of the objects involved in the collision. To calculate the final velocities, we can use the principles of conservation of momentum and kinetic energy.

Conservation of Momentum

The conservation of momentum principle in one-dimensional elastic collisions can be expressed as:

[ m_{1}v_{1i}+m_{2}v_{2i}=m_{1}v_{1f}+m_{2}v_{2f} ]

where:

(m_{1}) and (m_{2}) are the masses of the two objects
(v_{1i}) and (v_{2i}) are the initial velocities of the objects
(v_{1f}) and (v_{2f}) are the final velocities of the objects

Conservation of Kinetic Energy

Elastic collisions also conserve kinetic energy. The kinetic energy before and after the collision is equal:

[ \frac{1}{2}m_{1}v_{1i}^{2}+\frac{1}{2}m_{2}v_{2i}^{2}=\frac{1}{2}m_{1}v_{1f}^{2}+\frac{1}{2}m_{2}v_{2f}^{2} ]

Solving for Final Velocities

To find the final velocities, we can rewrite the conservation of momentum equation as:

[ v_{2f}=v_{1f}+v_{1i}-v_{2i} ]

and substitute it into the conservation of kinetic energy equation:

[ \frac{1}{2}m_{1}v_{1i}^{2}+\frac{1}{2}m_{2}(v_{1f}+v_{1i}-v_{2i})^{2}=\frac{1}{2}m_{1}v_{1f}^{2}+\frac{1}{2}m_{2}v_{2f}^{2} ]

This equation can be simplified and solved to find the final velocities (v_{1f}) and (v_{2f}).

Example

Consider a one-dimensional elastic collision between a 4 kg ball moving at 5 m/s and an initially at rest 4 kg ball. We want to find the final velocities of the balls after the collision.

Using the conservation of momentum equation:

[ 4(5)+4(0)=4v_{1f}+4v_{2f} ]

We get:

[ 20=4v_{1f}+4v_{2f} ]

Now, we can use the conservation of kinetic energy equation:

[ \frac{1}{2}(4)(5)^{2}+\frac{1}{2}(4)(0)^{2}=\frac{1}{2}(4)v_{1f}^{2}+\frac{1}{2}(4)v_{2f}^{2} ]

This simplifies to:

[ 100=2v_{1f}^{2}+2v_{2f}^{2} ]

Now, we can combine the two equations:

[ 20=4v_{1f}+4v_{2f} ]

[ 100=2v_{1f}^{2}+2v_{2f}^{2} ]

We can solve this system of equations to find the final velocities:

[ v_{1f}=-3.00,\text{m/s} ] [ v_{2f}=1.00,\text{m/s} ]

Discussion

The result of this example is intuitively reasonable. A small object strikes a larger one at rest and bounces backward. The larger one is knocked forward, but with a low speed. The equations for conservation of momentum and internal kinetic energy as written above can be used to describe any one-dimensional collision, and they can be extended to more objects if needed.

Description

This quiz covers the principles of conservation of momentum and kinetic energy in one-dimensional elastic collisions, and how to calculate the final velocities of objects involved in the collision. It includes examples and equations to solve for the final velocities.

Elastic Collisions in One Dimension: Velocity Calculations

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Questions and Answers

What is the equation that relates the initial and final velocities in an elastic collision?

In an elastic collision, what is conserved?

What is the result of the example collision in the text?

How can the equations for conservation of momentum and kinetic energy be extended?

What is the final velocity of the 4 kg ball that was initially at rest in the example?

What is the purpose of the conservation of momentum equation in the example?

What is the relationship between the momentum and kinetic energy of a system before and after a one-dimensional elastic collision?

What is the equation that represents the conservation of momentum principle in one-dimensional elastic collisions?

What is the purpose of using the principles of conservation of momentum and kinetic energy in one-dimensional elastic collisions?

What is true about the kinetic energy of the system in one-dimensional elastic collisions?

What is required to calculate the final velocities of the objects involved in a one-dimensional elastic collision?

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