3-6: 2D Collisions
Introduction
In the last two classes, we have learned how to analyze collisions in 1D situations. In today’s class, we will expand our understanding of collisions and how to apply Conservation of Momentum by looking at 2D situations.
After today’s class you will be able to:
- Apply Conservation of Momentum to 2D collisions
- Define the Line of Impact (LOI) and Plane of Contact (POC) for a 2D collision
- Describe how LOI and POC relate to analyzing a 2D collision
2D Perfectly Inelastic Collisions
Up to this point, we have defined conservation of momentum for 1D collisions. However, many real world collisions don’t occur along a straight line. How do we use conservation of momentum to understand 2D collisions like the collision of two billiard balls or two cars crashing into one another?
We can use the same basic principle we used for 1D collisions, but because momentum is a vector, we can write conservation of momentum equations in both the x and y directions. For a perfectly inelastic collision in 2D, the colliding objects stick together after the collision and have a common final velocity.
$m_{1}v_{1x} + m_{2}v_{2x} = (m_{1} + m_{2})v_{x}^{\prime}$
$m_{1}v_{1y} + m_{2}v_{2y} = (m_{1} + m_{2})v_{y}^{\prime}$
Example Problem
Mabel the dog jumps onto a raft. Assuming the dog and raft stay together, what is their final velocity (magnitude and direction)?
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Collisions in Multiple Dimensions Textbook Pages
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2D Perfectly Elastic Collisions
Unlike a perfectly inelastic collision, when we have a perfectly elastic collision in 2D, the colliding objects do not stick together after the collision. Just like in all collisions, however, total momentum is still the same before and after the collision:
$m_{1}v_{1x} + m_{2}v_{2x} = m_{1}v_{1x}^{\prime} + m_{2}v_{2x}^{\prime}$
$m_{1}v_{1y} + m_{2}v_{2y} = m_{1}v_{1y}^{\prime} + m_{2}v_{2y}^{\prime}$
Kinetic energy is also conserved in perfectly elastic collisions, but remember that energy is not a vector, so we will only have one equation to work with here:
$\frac{1}{2}m_{1}v_{1}^{2} + \frac{1}{2}m_{2}v_{2}^{2} = \frac{1}{2}m_{1}(v_{1}^{\prime})^{2} + \frac{1}{2}m_{2}(v_{2}^{\prime})^{2}$
Example Problem
Two hockey pucks collide. Puck A is initially moving to the right at 5 m/s when it collides with puck B. After the collision, puck A moves at 2 m/s at an angle of 30 degrees from its original direction. What is the speed and direction of puck B after the collision?
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2D Inelastic Collisions
If a collision is inelastic, but not perfectly inelastic, then the colliding objects do not stick together afterwards, but some kinetic energy is lost to other forms of energy such as sound, thermal energy, etc. Most real world collisions fall into this category.
In the last lesson, we learned that the coefficient of restitution (e) relates the relative velocities of the objects before and after the collision.
$e = \frac{-(v_{1}^{\prime} -v_{2}^{\prime})}{v_{1} - v_{2}}$
To analyze 2D inelastic solutions, we can use a method that relies on knowing some information about the plane of contactand line of action of the collision.
Watch the following video to see how to solve these types of problems:
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Example Problem:
The cue ball A is given an initial speed of 5 m/s. It makes direct perfectly elastic impact with ball B, giving ball B a speed of 5 m/s. Ball B then makes contact with cushion C (e = 0.6). Each ball has a mass of 0.4 kg. Neglect friction and the size of each ball.
Determine the speed of ball B and the angle θ after ball B hits cushion C.
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Answer these concept questions on 2D collisions.
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Solve this 2D inelastic collision problem:
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Summary
- 2D COM - 2 equations for components for conservation of momentum
- LOI - Line of impact - velocity components related by COR equation
- POC - Plane of contact - velocity components remain the same
Class Demos
