SCN3797-ABED_1: 5.1 Conservation in Non-Linear Collisions

What is the difference between a linear and non-linear interaction? To view an example of a non-linear collision, watch this animation.

In the previous lesson you learned about momentum and the law of conservation of momentum in the context of one-dimensional collisions: for any isolated system, the total momentum does not change. In a collision, momentum is conserved; the total momentum before the collision is equal to the total momentum after the collision.

Is the same true of two-dimensional collisions?

Intersections are often the location of many 2-dimensional collisions.
Image by Adrian Malec from Pixabay

Need to Review Vector Addition?

Want a "vector refresher?" Go through the examples in the Vector Review page in your course.

Lab Simulation: Two-Dimensional Collisions

Use the Collision Lab to help you answer the following questions.

Problem

Is momentum conserved in two-dimensional collisions?

Procedure

Once the simulation is open, follow these steps:

Select the "advanced " tab from the top left of the screen
Select "2 dimensions" on the right-hand menu
Toggle "More Data" below the data table
Toggle the "show paths" from the menu on the right

Perform one two-dimensional collision; then complete the following table. To generate a new collision, press the "restart" button to the left of the data table.

Observations

Collision 1 (sample data)
Object	Mass (kg)	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mover mathcolor=¨#FFFFFF¨»«mi»v«/mi»«mo»§#8594;«/mo»«/mover»«mi mathcolor=¨#FFFFFF¨»i«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mover mathcolor=¨#FFFFFF¨»«mi»v«/mi»«mo»§#8594;«/mo»«/mover»«mi mathcolor=¨#FFFFFF¨»f«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«mo mathcolor=¨#FFFFFF¨»§#8710;«/mo»«mover mathcolor=¨#FFFFFF¨»«mi»v«/mi»«mo»§#8594;«/mo»«/mover»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mover mathcolor=¨#FFFFFF¨»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mi mathcolor=¨#FFFFFF¨»i«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»k«/mi»«mi mathcolor=¨#FFFFFF¨»g«/mi»«mo mathcolor=¨#FFFFFF¨»§#183;«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mover mathcolor=¨#FFFFFF¨»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mi mathcolor=¨#FFFFFF¨»f«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»k«/mi»«mi mathcolor=¨#FFFFFF¨»g«/mi»«mo mathcolor=¨#FFFFFF¨»§#183;«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«mo mathcolor=¨#FFFFFF¨»§#8710;«/mo»«msub»«mover mathcolor=¨#FFFFFF¨»«mi»p«/mi»«mo»§#8594;«/mo»«/mover»«mi mathcolor=¨#FFFFFF¨»i«/mi»«/msub»«mspace linebreak=¨newline¨»«/mspace»«mo mathcolor=¨#FFFFFF¨»(«/mo»«mi mathcolor=¨#FFFFFF¨»k«/mi»«mi mathcolor=¨#FFFFFF¨»g«/mi»«mo mathcolor=¨#FFFFFF¨»§#183;«/mo»«mi mathcolor=¨#FFFFFF¨»m«/mi»«mo mathcolor=¨#FFFFFF¨»/«/mo»«mi mathcolor=¨#FFFFFF¨»s«/mi»«mo mathcolor=¨#FFFFFF¨»)«/mo»«/mstyle»«/math»
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Analysis

Questions:

How would you calculate the total momentum before and after a two-dimensional collision?
Would the same analysis that you used for one-dimensional situations work here?
According to your observations, is momentum conserved in a two-dimensional collision?

According to the data you collected for in the previous simulations and using the typical analysis performed for a one-dimensional collision, you should find that momentum in a two-dimensional collision is not conserved. However, this is contrary to the law of conservation of momentum, which means a new type of analysis must be used for two-dimensional collisions.

This new type of analysis is based on the following principles, which apply to two-dimensional interactions:

Momentum in the x-direction is conserved.
Momentum in the y-direction is conserved.

Using the data below, determine the total momentum before and after the collision using the analysis method that was just introduced to you.

Object	Mass (kg)	«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mstyle indentalign=¨center¨»«mover mathcolor=¨#FFFFFF¨»«mi»v«/mi»«mo»§#8594;«/mo»«/mover»«/mstyle»«mi mathcolor=¨#FFFFFF¨»i«/mi»«/msub»«/mstyle»«/math»		«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle indentalign=¨center¨»«msub»«mstyle indentalign=¨center¨»«mover mathcolor=¨#FFFFFF¨»«mi»v«/mi»«mo»§#8594;«/mo»«/mover»«/mstyle»«mi mathcolor=¨#FFFFFF¨»f«/mi»«/msub»«/mstyle»«/math»
Object	Mass (kg)	Magnitude (m/s)	Direction (degrees)	Magnitude (m/s)	Direction (degrees)
blue ball	5.00	8.00	0	2.80	69.51
green ball	5.00	0.0	0	7.50	−20.49

Questions:

What is the initial total momentum in the x direction?
What is the initial total momentum in the y direction?
What is the final total momentum in the x direction?
What is the final total momentum in the y direction?

On the next page is an example of how to describe vectors (such as momentum) using components and how to apply the law of conservation of momentum correctly to a two-dimensional collision.