042 Laboratory Four: Momentum

03 February 2009

Loosing your marbles

Introduction

Do NOT copy and paste this whole thing into your lab report. Write your report in your own words!

This laboratory explores the concepts of momentum and conservation of momentum.

Terminology: Large, shooter marbles are called taws. Small marbles are called ducks. What do you call marbles? What do you call shooter and player marbles? In this lab we will use only duck marbles.

Questions

• What does momentum mean?
• What does conservation of momentum mean?
• Can we show that momentum is conserved in simple systems?

Introduction

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 duck marble and another duck marble. In the third part you will repeat part two, but using a taw colliding with a duck.

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).
  $\stackrel{\rightharpoonup }{P}=m\stackrel{\rightharpoonup }{v}$
• 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.

Equipment

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

Part One: Conservation of Marble Momentum: Rolling Ducks

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.

Data tables [d] [t] | Data Analysis and Results | Data Display/Diagrams

Design your own. You decide how to best record and present the data you have gathered.

Part Two: Conservation of momentum in a duck-duck 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 rample. The momentum of the one duck rolling down the ramp before the collision should be equal to the sum of the momentums of the ducks after the collision.

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

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

$\begin{array}{}{\stackrel{\rightharpoonup }{P}}_{\text{before}}=\sum {\stackrel{\rightharpoonup }{P}}_{\text{after}}\\ m_{1_{\text{before}}}\times {\stackrel{\rightharpoonup }{v}}_{1_{\text{before}}}=m_{1_{\text{after}}}\times {\stackrel{\rightharpoonup }{v}}_{1_{\text{after}}}+m_{2_{\text{after}}}\times {\stackrel{\rightharpoonup }{v}}_{2_{\text{after}}}\end{array}$

The blue duck has a mass m_blue (m1) and the white duck that is hit on the track is mass m_white (m2) in the formula above.

In part two we measure all of the variables above and then plug the values into the equation above. If the left side is equal to the right side, momentum is conserved. If the left and right side are within 10% of each other, then the agreement is good enough. If the left side and right side are within 10% of each other, then we cannot say momentum is not conserved (watch the double negative!).

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

Overview

1. Find the mass of both of the ducks. Use two ducks close to the same mass if possible.
2. Rolling only one duck, the "inbound" m1 duck, measure the speed of the duck with the track empty. This is the velocity (speed) before the collision. Remember, the other duck is sitting still on the track with a velocity of zero centimeters per second.
3. Roll the one duck from the same spot so it collides with the second duck on the flat section of the track. Measure the speeds of each of the ducks after the collision.
4. Use the values obtained to plug into the equation above and determine if momentum before is the same as momentum after.

Details

The mass is measured using a balance beam scale.

To measure the speed accurately, we will roll the marble five times measuring the length of time for the marble to roll across the 30 cm flat section of track on the "second" ruler. This means measuring five time durations before the collision, and ten measurements for the durations after the collision.

Before: Inbound marble speed measurement procedure

1. Roll the inbound blue duck down the track by itself, releasing the duck from 0.0 cm at the top of the ramp track.
2. Measure the time for the blue duck 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 duck.
3. Repeat this five times to get the average time for the blue duck prior to being involved in the collision.

We measure the speed on the flat section. The 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.

To reduce the error, take five time measurements and use the average time. Instructional note: lower blocks yield slower marbles which improve velocity measurements. Typical a single roughly one to 1.5 cm high block is used.

Mean time t1 from table above: _________________

The speed of the blue marble is calculated from velocity = distance ÷ duration. For the above set-up, the calculation is velocity v = 30 cm ÷ mean time t1

Calculate the momentum of the inbound blue duck.

After: Outbound marble speeds measurement procedure

Now set up the ducks to collide.

Post-collision procedure

1. Place the blue m1 duck at 0.0 cm on the ramp track.
2. Place the white m2 duck 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.

The above will require making five time measurements of the blue and five of the white duck. Use these measurements to determine the mean time for each. The next two tables provide a place to record data.

Mean time t2 from table above: _________________

Mean time t3 from table above: _________________

Is the momentum of the inbound m1 duck equal to the sum of the momentums of the two ducks after the collision? How close are the results? Use the percentage change formula to determine the change in momentum:

$\text{Percentage change Δ\%}=\frac{(\text{sum of the momentums after}-\text{momentum before})}{\text{momentum before}}$

If the percentage change is less than 10%, then based on our very basic experiment we cannot rule out conservation of linear momentum.

The momentum after is not usually exactly equal to the momentum before. Was momentum gained or lost from before to after? Why do you think this happened?

Data Display | Diagrams

Optional and up to the student.

Analysis [a]

• What did you find – was momentum conserved?
• What is the percentage gain or loss in momentum?
• Where is the momentum coming from or going to– if anywhere?
• Using (momentum after − momentum before)/momentum before, is your result within the 10% uncertainty before we tend to expect in our laboratory measurements?

Conclusions [c]

Wrap up these two activities with an essay that addresses each of the two activities and 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? How large, on a percentage basis, was the gain or loss? Discuss anything unusual, new, ordifferent 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.

Optional extension: Use ducks of different sizes in part two. Gather data. Plot P_before versus P_after on an xy scattergraph.