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G9-4b. Solving Elastic Collision Problems Using the Center-of-Mass Method

Introduction

We'll use the facts below to calculate the final velocities of any two objects that collide with no loss of kinetic energy in one dimension.

• The center of mass of an object or system of objects is the one point that behaves as if the total mass of the system were located at that point.
• The total momentum of a system is equal to the mass of the system multiplied by the velocity of the center of mass (CM).
• For completely elastic collisions, the velocity at which an object in the system approaches the CM is equal to the opposite of the velocity at which that same object separates from the CM.

The following two examples will illustrate the method. We assume the direction of positive displacement is to the right. The velocities are the velocities with respect to the frame of reference of the laboratory (for example, an air track on which the collision occurs).

Example 1

See the diagram below for the masses and initial velocities. The goal is to find the velocities of both blocks after an elastic collision.

1. Find the velocity of the center of mass of the system of the two blocks.

Vcm = Pi / Msys

= (m1v1i + m2v2i) / Msys

= [(1.0 kg)(6.0 m/s) + (2.0 kg)(3.0 m/s)] / (3.0 kg)

= +4.0 m/s

1. Calculate how rapidly each object is approaching the CM.

v1,app = v1i - Vcm = +2.0 m/s

v2,app = v2i - Vcm = -1.0 m/s

1. Calculate the separation velocities from the CM as the opposites of the approach velocities.

v1,sep = -v1,app = -2.0 m/s

v2,sep = -v2,app = +1.0 m/s

1. Calculate the final velocities.

v1f = Vcm + v1,sep = +2.0 m/s

v2f = Vcm + v2,sep = +5.0 m/s

1. As a check, use the final velocities to calculate the final total momentum to see if it’s the same as the initial total momentum.

Pf = m1v1f + m2v2f

= (1.0 kg)(2.0 m/s) + (2.0 kg)(5.0 m/s)

= 12.0 kgm/s

= Pi

Simplifying the Method

We can combine the formulas in steps 2 to 4 above to simplify the method.

v1f = Vcm + v1,sep

= Vcm - v1,app

= Vcm - (v1i - Vcm)

= 2Vcm - v1i

In a similar way, we find that

v2f = 2Vcm - v2i

Example 2

Use the simplified formulas above to determine the final velocities of the objects in the elastic collision below.

1. Calculate the center of mass velocity.

1. Calculate the final velocities.

1. Check the result.

Summaries of the Methods

The Long Method

1. Calculate the velocity of the CM using .
2. Calculate the approach velocity of each mass as the difference of the initial lab velocity and the CM velocity.
3. Calculate the separation velocities as the negatives of the approach velocities.
4. Calculate the final lab velocity of each mass as the sum of the CM velocity and the separation velocity.
5. Check that the final total momentum equals the initial total momentum.

The Short Method

1. Calculate the velocity of the CM using .
2. Calculate the final lab velocities using and .
3. Check that the final total momentum equals the initial total momentum.

A Graphical Method

A velocity vs. time graph can be used to illustrate how the method above works. The graph below is for Example1. The red and blue lines represent the red and blue objects respectively, and the green line represents the center of mass. Note that during the collision, the initial velocities of the objects are reflected vertically about the center of mass velocity.

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