COLLEGE PHYSICS LABORATORY
PH 141
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Collisions and the Conservation of Momentum

Introduction

The conservation of linear momentum means that any change in the momentum of a system must be accounted for by a net transfer of momentum to or from the system. Momentum is a vector quantity, usually identified by the symbol p (the boldface type indicates vector quantity). Conservation of linear momentum can be expressed as:

pf - pi = pin - pout

If there are no external forces which effect momentum transfers or the duration of the interaction is so brief that no momentum is transferred to the system, then this relationship is simplified.

pf - pi = 0

This could also be rewritten: pf = pi

For a system of many objects:  Spij = Spfj

(The subscript "i" means initial, the subscript "f" means final, the subscript "j" indicates which of many objects which make up the system. The "S" symbol means "sum of," i. e. add together the contributions of all objects.)

For a collision between two objects, the principle simplifies to the following equation:

m1vi1 + m2vi2 = m1vf1 + m2vf2

As with any other vector equation, this equation can be considered to represent the combination of independent motions in the x- and y-directions (however they may be defined):

m1vi1x + m2vi2x = m1vf1x + m2vf2x

and

m1vi1y + m2vi2y = m1vf1y + m2vf2y



    In order to apply the principle of momentum conservation, one must first specify the system under consideration. Secondly, one must specify what moments in time one wishes to compare. According to the principle, the momentum of a system remains the same as long as there is no introduction of momentum from outside the system nor net loss of momentum from the system. However to apply the mathematical relationship, one must specify two points in time to compare.

INVESTIGATIONS

    The purpose of these investigations is to gain experience in applying the principle of Conservation of Momentum to actual events. Different cases will be examined. First, collisions between hard objects will be investigated. Two cases will be explored: collisions having equal masses and collisions between objects having unequal masses. Later, collisions between "sticky" or "soft" objects will be investigated.

Collisions I

Collisions between hard objects of equal mass

Preparation:
1. Adjust the ramp to some height above the sandbox. Adjust the ramp so that the ball projects out from the ramp horizontally.
2. Adjust the screw in front of the end of the ramp so that the ball that rests on the screw is level with the ball that projects off of the ramp. (A good way to check if the two balls are level is to initiate a collision by rolling one ball down the ramp and listening to hear if the two balls strike the sand at the same time. Why is this evidence that the balls strike on a level horizontal plane?)
3. Identify a point in the sand which lies directly below the point of collision.
4. Release the ball down the ramp and mark where the ball lands with no collision.Measure from the point directly below the (future) collision point to the point of impact. Do this a couple times. Does the ball land in the same spot consistently? If not, you need to improve your technique.
5. Mark the path of projection with a line.

Exploration:
Perform several practice collisions in order to get a feel for where the balls will land.
Plan a series of 6 collisions such that landing position of the target ball extends over as wide a range as possible.

Method: For each collision
1. Clearly identify impact points as being due either to the target ball or the projected ball.
2. Measure distance to the impact point of each ball from the collision point.
3. Measure the angle between the path of projection (the path of the projected ball when there is no collision and the direction to the point of impact.

 Results and Analysis: 1. Compile your results in a table.
2. For any two collisions of your choice (hint: glancing collisions usually provide better results), demonstrate that the principle of momentum conservation is valid: a. Assume that the distance between the impact and collision points represents the velocity of a ball. Why is this a reasonable assumption?
b. Determine the x- and y-components of the motion for the target ball and the projected ball.
c. Verify mathematically that momentum was conserved in both the x- and y-directions independently. 3. Transfer your results to a diagram. a. On a piece of paper draw to scale two points which will represent the point of collision and the impact point of the projectile when there is no collision.
b. Using the same scale, indicate the points where the target and projectile balls struck for each of your collisions.
c. Draw lines connecting the impact points of the target and projected balls. 4. Interpret - is there any apparent pattern to the positions of the points of impact?
 
Collisions II

Collisions between hard objects of unequal mass

Preparation:
1. Adjust the ramp to some height above the sandbox. Adjust the ramp so that the ball projects out from the ramp horizontally.
2. Adjust the screw in front of the end of the ramp so that the ball that rests on the screw is level with the ball that projects off of the ramp. (A good way to check if the two balls are level is to initiate a collision by rolling one ball down the ramp and listening to hear if the two balls strike the sand at the same time. Why is this evidence that the balls strike on a level horizontal plane?)
3. Identify a point in the sand which lies directly below the point of collision.
4. Release the ball down the ramp and mark where the ball lands with no collision.Measure from the point directly below the (future) collision point to the point of impact. Do this a couple times. Does the ball land in the same spot consistently? If not, you need to improve your technique.
5. Mark the path of projection with a line.
6. Measure the masses of your two objects.

Exploration:
Use the lighter ball as the target ball and the heavier ball as the projected ball.
Perform several practice collisions in order to get a feel for where the balls will land.
Plan a series of 6 collisions such that landing position of the target ball extends over as wide a range as possible.

Method: For each collision
1. Clearly identify impact points as being due either to the target ball or the projected ball.
2. Measure distance to the impact point of each ball from the collision point.
3. Measure the angle between the path of projection (the path of the projected ball when there is no collision and the direction to the point of impact.

 Results and Analysis: 1. Compile your results in a table.
2. For any two collisions of your choice (hint: glancing collisions usually provide better results), demonstrate that the principle of momentum conservation is valid: a. Assume that the distance between the impact and collision points represents the velocity of a ball. Why is this a reasonable assumption?
b. Determine the x- and y-components of the motion for the target ball and the projected ball.
c. Verify mathematically that momentum was conserved in both the x- and y-directions independently. 3. Transfer your results to a diagram. a. On a piece of paper draw to scale two points which will represent the point of collision and the impact point of the projectile when there is no collision.
b. Using the same scale, indicate the points where the target and projectile balls struck for each of your collisions.
c. Draw lines connecting the impact points of the target and projected balls. 4. Interpret - is there any apparent pattern to the positions of the points of impact?