# 042 Laboratory Four: Momentum

## Questions

• What does momentum mean?
• What does conservation of momentum mean?
• Can we show that momentum is conserved in simple systems?
• Is there a linear relationship between the momentum before a collision and the momentum after a collision? Is the momentum conserved?

### Introduction

This laboratory explores the concepts of momentum and conservation of momentum. Existing theory asserts that momentum is conserved. In the first part of this two-part laboratory you will explore qualitatively the conservation of momentum. In the second part you will calculate the momentum before a collision and the momentum after a collision of a marble and another marble. These collisions will be run for three pairs of marbles, each pair will be similar in mass.

Terminology: Large, shooter marbles are called taws. Small marbles are called marbles. marbles come in different sizes. What do you call marbles? What do you call shooter and player marbles? "Loosing" means "to let loose" as in "to release." Loosing is not the same as "losing" your marbles. "Losing" means to become misplaced, to become lost. Do not lose your marbles, loose them! In physics:

• Momentum is the mass (grams) multiplied by the velocity (cm/s). The letter P is used for momemtum, m is used for mass, and v is used for velocity (speed).
$\stackrel{⇀}{P}=m\stackrel{⇀}{v}$
• Conservation means "stays the same." Usually this means, "the momentum after an event is the same as the momentum before an event." For this lab the "event" is a collision between marbles.
• duck is the term for a small marble.
• taw is the term for a large marble.

### Equipment

• marbles
• rulers
• stopwatch
• wood block or other support
• tape

## Part One: Conservation of Marble Momentum: Rolling marbles In part one we explore a simple system. Five marbles sit touching each other on the flat portion of a marble track. The marble track is made of two plastic rulers with grooves to guide the marbles. One or more marbles are released from an elevated end of the track. ### Procedure for part one

1. Release one marble. How many marbles are ejected ("kicked out") from the group?
2. Release two marbles. How many marbles are ejected from the group?
3. Repeat for three, four, five... marbles.
4. How is the number in related to the number out?
5. Release one marble from half-way up the ramp. Is the inbound marble fast or slow? Is the ejected marble fast or slow?
6. Send a marble in at high speed. Is the ejected marble fast or slow?
7. How is the speed (velocity) in related to the speed (velocity) out?

As you work on the above questions, experiment. Play with the marbles. How to the marbles know what to do? How does a marble know whether to go or to stay? How do the marbles count? Just how smart is a marble? Play gently – marbles can and do break – but do play.

Instructional option: Have the students work in groups for an hour to try to verify conservation of momentum (see part two) using the apparatus of part one, stopwatches, and mass balances. Then the groups present their findings to the class.

### [s] Sketch or drawing

Design your own. You decide how to best present marbles in = marbles out, speed in = speed out in a drawing or sketch.

## Part Two: Conservation of momentum in a collision

The momentum p is defined as the mass multiplied by the velocity (speed). Both momentum and velocity have directions associated with them, both are vector quantities. This means they are usually written with an arrow on top of the symbol for them. Marbles have a mass, their velocity is a speed in a particular direction. The tracks keep the marbles moving in the same single direction. In the world of science this is a one-dimensional model and keeps the mathematics simpler.

### Part two introduction

Momentum is said to be conserved. This means that the momentum before an event should be equal to the momentum after an event.

In part two the event is a collision between two marbles. One marble at rest is hit by another marble rolling down the ramp. The momentum of the one marble rolling down the ramp before the collision should be equal to the sum of the momentums of the marbles after the collision.

The marble coming into the collision is called the "inbound" marble in this laboratory. To keep the marbles straight, this lab will refer to the inbound marble as the blue marble and the marble that is sitting still on the track at the start as the white marble. Your marbles may be different in color!

Said "mathematically," the momentum before is equal to the sum of the momentums after is written:

$\begin{array}{}{\stackrel{⇀}{P}}_{\text{before}}=\sum {\stackrel{⇀}{P}}_{\text{after}}\\ {m}_{{1}_{\text{before}}}×{\stackrel{⇀}{v}}_{{1}_{\text{before}}}={m}_{{1}_{\text{after}}}×{\stackrel{⇀}{v}}_{{1}_{\text{after}}}+{m}_{{2}_{\text{after}}}×{\stackrel{⇀}{v}}_{{2}_{\text{after}}}\end{array}$

The blue marble has a mass mblue (m1) and the white marble that is hit on the track is mass mwhite (m2) in the formula above.

This collision will be repeated for marbles of three different sizes. Each collision will use marbles that are close to the same size.

#### Overview

1. Find the mass of both of the marbles. Use two marbles close to the same mass if possible. Start with the smallest marbles.
2. Rolling only one marble, the "inbound" m1 marble, measure the speed of the marble with the track empty. This is the velocity (speed) before the collision. Remember, the other marble is sitting still on the track with a velocity of zero centimeters per second.
3. Roll the one marble from the same spot so it collides with the second marble on the flat section of the track. Measure the speeds of each of the marbles after the collision.

#### Details

The mass is measured using a balance beam scale.

Instructional notes: The measurement process is not obvious. The instructor should perform a single collision, measuring the masses of the marbles and the velocities before and after, filling out a sample set of tables.

#### Before: Inbound marble speed measurement procedure for a pair "regular" duck marbles.

1. Roll the inbound blue marble down the track by itself, releasing the marble from 0.0 cm at the top of the ramp track.
2. Measure the time for the blue marble to cover the 30.0 cm along the flat ruler. The two marbles below show the distance over which the measure the time for the blue marble. We measure the speed on the flat section. On the slope the marble is accelerating. We only want to know the speed of the marble at the bottom of the slope. The speed at the bottom of the slope is the speed at which the blue marble will collide with the white marble.

Calculate the momentum of the inbound blue marble.

Table 1: Momentum before collision
mass × velocity = momentum
mass m1 blue marble (g) distance d1 for blue marble(cm) time t1 for blue marble (s) momentum p1 blue marble (g cm/s)
× ÷ =

#### After: Outbound marble speeds measurement procedure

Now set up the marbles to collide. #### Post-collision procedure

1. Place the blue m1 marble at 0.0 cm on the ramp track.
2. Place the white m2 marble on the flat track at 0.0 cm.
3. In the image above m1 is on the right, m2 is on the left.
4. Run the collision. Both marbles will roll off the track.
5. Speed of the white m2 marble after collision: Rerun the collision timing the duration (time) for the white marble to travel 30 cm. Repeat the collision four more times, measuring the duration for the white marble to travel 30 cm to the end of the track.
6. Speed of the blue m1 marble after collision: Rerun the collision timing the duration (time) for the BLUE marble to travel 30 cm. Repeat the collision four more times, measuring the duration for the blue marble to travel 30 cm to the end of the track.
Table 2: Momentum after collision
mass × velocity = momentum Sum of P2 + P3
mass m1 blue marble (g) distance d2 for m1 (cm) time t2 for m1 after (s) momentum P2 after (g cm/s)
× ÷ =
mass m2 white marble (g) distance d3 for m2 after (cm) time t3 for m2 after (s) momentum P3 white marble after (g cm/s)
× ÷ =

Momentum before, P1 = _____________
Momentum after, P2 + P3 = _____________

### Small (or tiny) sized duck marble [sinsai]

Repeat the calculations above for a "normal" duck marble

Table 3: Momentum before collision
mass × velocity = momentum
mass m1 marble (g) distance d1 for m1 (cm) time t1 for marble (s) momentum p1 inbound marble (g cm/s)
× ÷ =
Table 4: Momentum after collision
mass × velocity = momentum Sum of P2 + P3
mass m1 marble (g) distance d2 for m1 (cm) time t2 for m1 after (s) momentum P2 after (g cm/s)
× ÷ =
mass m2 marble (g) distance d3 for m2 after (cm) time t3 for m2 after (s) momentum P3 marble after (g cm/s)
× ÷ =

Momentum before, P1 = _____________
Momentum after, P2 + P3 = _____________

### Taw or shooter sized marble [palas]

Repeat the calculations above for the largest marble, a taw or shooter marble.

Table 5: Momentum before collision
mass × velocity = momentum
mass m1 marble (g) distance d1 for m1 (cm) time t1 for marble (s) momentum p1 inbound marble (g cm/s)
× ÷ =
Table 6: Momentum after collision
mass × velocity = momentum Sum of P2 + P3
mass m1 marble (g) distance d2 for m1 (cm) time t2 for m1 after (s) momentum P2 after (g cm/s)
× ÷ =
mass m2 marble (g) distance d3 for m2 after (cm) time t3 for m2 after (s) momentum P3 white marble after (g cm/s)
× ÷ =

Momentum before, P1 = _____________
Momentum after, P2 + P3 = _____________

## Procedure | Data tables | Data Analysis and Results [d] [t]

Transfer data from above to the following table. Include this table in your report. When making your chart do NOT select the first column. Select only the second [x] and third [y] columns.

Table 7: Momentums
Marble momentum P1 before (g cm/s) [x] momentum P2 + P3 after (g cm/s) [y]
no marble 0 0
tiny duck marble
"regular" duck marble
taw marble

Instructor discussion point: What happens if the momentum before is zero? What will be the momentum after? If a nothing with no mass and no velocity collides with a marble at rest, will the marble move? Nothing in, nothing out. Include (0, 0) as a value in your table above.

## Graph [g]

Make an xy scatter graph with the momentum before the collision on the x axis and the momentum after on the y axis.

## Conclusions [c]

Wrap up these two activities with an essay that addresses part one and part two including the results you observed and measured. Comment on whether the hypotheses held for your team. Was momentum conserved in parts one and two? If momentum was lost or gained, why might it have been lost or gained? Discuss anything unusual, new, or different you encountered. Discuss what the conservation of momentum and energy means for you in light of the above activities. Be thorough and complete. Use correct grammar and spelling.

[Notes from the field for instructors: In this laboratory we explore conservation of linear momentum. Another momentum that is conserved is angular momentum. Angular momentum is the momentum of spinning. Spinning objects tend to continue to spin. Objects that are not spinning tend to remain at rest– to not spin. Think of a child's toy top. In the experiments above we considered only linear momentum, but the marbles are spinning as they move on the track. In part two a spinning m1 marble hits a non-spinning m2 marble. The m1 marble loses speed and thus spin, the m2 marble goes from not spinning (sitting still on the track) to spinning very quickly. These changes in spin momentum are related to why linear momentum is consistently "lost" in these collisions.

Where linear momentum is p = mv, the angular momentum L = Iω where I = 0.4mr² and ω = v/r. Thus the angular momentum of a marble is L = 0.4mrv. One cannot just add all the momentums and hope for the best: the units are different. Ultimately one has to retreat to an energy position noting that the potential energy must appear as both linear and rotational kinetic energy in both of the marbles post-collision, along with losses to friction, sound, and any heat produced in the collision.

The thought occured as to what to try to reduce the impact of external torque exerted by the track. One idea was to lubricate the ruler track with some form of greaseless lubricant such as WD-40®.

WD40 was tried. The first complication is the tape no longer holds the tracks in place. This problem proved rather insurmountable. In addition, WD40 wound up everywhere - on hands, table tops, soaked into paper that slid into the WD40. Would need a greaseless lubricant. Even, the loss of taping ability would remain fatal.

Why not simply use pucks on an air table? Two key reasons. The puck and air table are unfamiliar to students - this raises the probability that the students will, in their own minds, see the whole thing as magic. Another mysterious thing in the modern world. Secondily, the lab should be as reproducible as possible by any teacher in the nation. Part one requires nothing more than what an instructor on an atoll might be able to get their hands on. Part two adds only one unlikely element - a stop watch.

Marbles on a track are very complex!]

Optional extension: Use marbles of different sizes in part two. Gather data. Plot Pbefore versus Pafter on an xy scattergraph.