PHY 100

THE PHYSICS OF COLLISIONS

In this lab we will investigate the physics of collisions. The two measurable quantities which we consider in collisions are linear (translational) momentum and energy. Colli-sions can occur elastically or inelastically; collisions which occur elastically are called elastic collisions and those which occur inelastically are called inelastic collisions. The difference between the two types of collisions is the effect the collision has on the total energy of the sys-tem, where the system is defined as the collection of objects involved in the collision.

Theory and Experimental Formulae:

Linear (Translational) Momentum:

Momentum is one of the most important and pervasive concepts in all of physics. There are two general types of momentum: linear (translational) momentum and angular momentum. We will only consider linear momentum in this lab and leave the topic of angular momentum for later in the class. There are several ways to interpret momentum, but there is one interpretation which helps justify the form of the equation which defines it.

If you get hit by a baseball traveling at a certain speed, it will hurt more than a Ping-Pong ball traveling at the same speed or a baseball traveling at a slower speed. This is because the faster moving baseball has more of something we call linear (translational) momentum. Since the main difference between the baseball and the Ping-Pong ball is mass, momentum must depend on mass; on the other hand, gently tossing a baseball does not hurt, so momentum must also de-pend on velocity. It should then seem reasonable that momentum is defined as:

(1)

Notice that since velocity is a vector, so too is linear momentum. Recall that a vector has a magnitude and a direction and can also be expressed as a summation of its components in each direction of the coordinate system. The component form of linear momentum can then be found by decomposing the velocity and momentum vectors in (1) into their constituent components in the desired coordinate system, typically the Cartesian coordinate system. For a rigid body in mo-tion in a three-dimensional Cartesian coordinate system, the component form of linear momen-tum is then given by:

(2)

Comparing both sides of (2) we see that, as one might expect given the form of equation (1), the components of linear momentum in each direction are given by:

We can also interpret linear momentum as a measure of how difficult it is to stop an ob-ject from moving. Newton’s Second Law of Motion states that the time rate of change of an object’s linear momentum () is equal to, and in the same direction as, the constant net force () acting on it, or symbolically (do not forget that momentum is a vector quantity!):

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(3)

In other words, applying a constant force () to an object causes the object’s momentum to change in the direction of the force () and at a rate proportional to its magnitude. Therefore, if we want to change the momentum of an object by an amount in a specific time period, we can then use Newton’s Second Law to determine the required force ().

Now consider applying a constant force () to an object continuously for some amount of time (). A natural question to ask is what effect this will have on the object’s momentum and, consequently, its velocity? We can answer these questions by first rearranging Newton’s Second Law and then using equation (1) to determine what effect a change in an object’s momentum has on its velocity.

(4)

(5)

As an example, imagine you are driving your car on the freeway at some initial velocity (). Consider the amount of time () it takes to bring your car to a stop (by slamming on the brakes. Since the brakes on your car can only exert some maximum force ()*, we see from equa-tion (5) that:

(6)

As one would expect, we see that the faster you are going and/or the heavier your car is (and hence the more momentum you possess), the longer it takes for you to come to a stop. We also see that the larger the force () (i.e. the stronger your brakes are), the faster you can stop your car. Notice that since the time () and the mass () must both be positive, either or must be nega-tive. In either case, this means that the forcemust be in the opposite direction as the initial veloc-ity (). Although it is obvious in this simple example that the force must oppose the velocity of the car to slow it down, in more complicated situations, facts such as these may not be so obvi-ous. This highlights an important aspect of physics: if you use the correct equations and are careful with the math, you need not have an intuitive expectation about the result.

Energy:

An object’s energy is a measure of how much work it can do on an external physical system and is defined as the sum of both its potential energy (U) and its kinetic energy

* * More precisely, the brakes exert a force () on the wheel and the resulting torque () opposes the rotation of the wheels, slowing the car down. However, the ultimate effect is equivalent to a force () opposing the forward mo -tion of the car, so ignoring the details, it is this effective force which we will consider here.

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(KE). Potential energy exists in many forms, such as gravitational, elastic, chemical, and elec -trostatic, among others. Kinetic energy is energy associated with movement and can arise from translational, rotational, and/or vibrational motion. The total energy of an object is then given by:

Conservation of Linear Momentum:

Conservation of linear momentum requires the momentum of the system to re-main the same before and after the collision. Calling and the initial and final momentum of the system, respectively, conservation of linear momentum for a system of n objects can be ex-pressed mathematically as:

(7)

Notice that the velocities of the objects, and hence their momentum, are vectors. Recall that when dealing with vectors, we must consider the component in each direction separately. Therefore, conservation of linear momentum, as expressed in equation (7), requires the following three equations to each be true simultaneously:

In other words, for conservation of linear momentum to be satisfied, linear momentum must be conserved in each direction separately.

Conservation of energy:

Conservation of energy means that the total energy of the system (i.e., the com-bined energy of all n objects) remains constant before and after the collision. However, this does not mean that all, or even any, of the objects will individually possess the same amount of energy before and after the collision. An object may transfer some or all of its energy to another object in the system during a collision, but if energy is conserved, the sum of their energies will be un-affected by the collision. Calling and the initial and final energy of the system, respectively, con-servation of energy for a system of n objects is expressed mathematically as:

(8)

Elastic and Inelastic Collisions:

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Collisions in which the objects bounce off each other without deforming or losing energy are called elastic. In an elastic collision, both linear momentum and energy are conserved. Col-lisions in which the total energy of the system is not conserved are called inelastic. Of course, energy is neither created nor destroyed in an inelastic collision, some is simply lost to the sur-rounding environment. This typically occurs if the collision produces sound, generates heat, or causes the objects to stick together and/or deform. We will experimentally verify this by calcu-lating the initial and final energy of the system, using the same equations derived for elastic col-lisions, and showing that energy was lost in the collision. However, like elastic collisions, linear momentum is still conserved in an inelastic collision. Therefore, the equations derived for con-servation of linear momentum in an elastic collision apply also to inelastic collisions and need not be derived again.

Although it is nearly impossible to produce a perfectly elastic collision and nearly all col-lisions produce at least some sound or heat, we will perform collisions that lose such a small amount of energy that they are indistinguishable from elastic collisions. Therefore, we will con-sider these collisions to be perfectly elastic for our purposes.

Experimental Apparatus:

To experimentally verify the properties of collisions, we will perform various collisions on the PASCO Dynamics Track. For simplicity, we will perform collisions between only two ob-jects and restrict their motion to one-dimension, since the same principles apply. Our two objects will be PASCars and we will vary their mass using the PASCar weights (one has a blue dot and one has a red dot to distinguish them from each other). Each cart will be attached to a rotary mo-tion sensor (RMS) so we can measure its velocity before and after the collision (i.e. the initial and final velocities of each cart). We can then calculate the momentum and en-ergy of the two carts before and after the collision and determine if momentum and/or energy were conserved.

A cart-string adapter is used to attach each cart to an RMS and each RMS is mounted to the same end of the track on an IDS Mount Accessory (as shown in the diagram below). Make sure that the RMS at-tached to the cart closest to the RMSs (cart 1 in the diagram) has its yellow cable connected to Input #1 and its black cable connected to Input #4 of the PASCO 850 Interface. The RMS at -tached to the other cart (cart 2) should have its yellow cable connected to Input #3 and its black cable connected to Input #2 of the interface. There should also be two pulleys mounted on IDS Track Pulley Brackets at the opposite end of the track as the RMSs, one on each side (as shown in the diagram).

To attach a cart to an RMS, first place the slot of a cart-string adapter over the top lip

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of the cart and tighten the bottom screw. If not already connected, wrap string around the rotary motion sensor’s largest pulley and the pulley at the other end of the track. Then wrap both ends of the string around the top screw on the adapter and tighten it. As you tighten the screw, the strings will tend to rotate, so try to hold them in place so the string appears to be a straight line, as shown in the diagram on the left. Adjust the height of the rotary motion sensor and pulley so that the top string is parallel to the track and free from obstruction (i.e. it does not rub on any-thing).

If the string is attached too tightly, the pulleys will add substantial friction and the cart will not move freely. If it is too loose, the string may either slip on the pulley as the cart moves, or even worse, not rotate the pulleys at all. The easiest way to produce the proper tension in the string is to adjust the horizontal location of the IDS Track Pulley Bracket until the string is just tight enough to prevent the pulley from slipping.

Experiment:

To experimentally verify the properties of collisions, we will perform various collisions on the PASCO Dynamics Track. For simplicity, we only perform collisions between two objects moving in one-dimension, since the same principles apply. Since there are only two objects, equation (7) describing conservation of linear momentum reduces to:

(9)

Since we are restricting motion of the two objects to one-dimension, we need not con-sider linear momentum as a full three-dimensional vector; we only need to consider the magni-tude of the velocity and whether it is positive or negative. Therefore, we can drop the vector no-tation in (9) and the equation we will use to determine if linear momentum is conserved is (re-member the velocities can be positive or negative!):

(10)

Similarly for energy, because there are only two objects, the equation for conservation of energy reduces to:

With movement confined to a horizontal track, neither cart will experience a change in gravitational potential energy during a collision. Since there is no other type of potential energy to consider in our system, the initial and final potential energy of each cart will be the same (i.e. and), causing these terms to cancel out. Another consequence of restricting the motion of the carts to one dimension is that the only type of kinetic energy present in our system will be from linear motion, which is given by. Inserting this in for the kinetic energy of each cart and cancel-

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ing the potential energy terms from both sides yields the following equation for conservation of energy:

(11)

We will use equations (10) and (11) for both elastic and inelastic collisions when consid-ering the momentum and energy of the system, respectively.

Experimental Procedure:

The collisions we will perform will essentially be the same for both elastic and inelastic collisions; we will simply change the orientation of the carts to cause the desired type of colli-sion. Collisions will occur between two PASCars on the Dynamics Track and each cart will be attached to a rotary motion sensor to record its velocity before and after the collision. Each cart has one end with three small magnets (each with the same polarity) and one end with a strip of Velcro. The blue carts have one side of the Velcro and the red carts have the other. Therefore, two blue carts will not stick together, nor will two red carts. Only a red cart and a blue cart can be made to stick together, which is why you have one of each.

At each station you will find two weights, one for each cart, which weigh approximately 250g each; the one with a blue dot is for the blue cart and the one with a red dot is for the red cart. We will adjust the weight of the carts throughout the course of the experiment by adding or removing these weights. Once we have the apparatus arranged for a particular type of collision and have adjusted the tension in the string, we want to avoid detaching the carts from the appara-tus each time we add or remove a weight and need to know their new mass. The easiest way to accomplish this is to measure and record the mass of the carts, the two weights, and the cart‒string adapter separately before beginning the experiment and simply add the appropriate masses together for a particular arrangement.

We will perform three distinct elastic collisions by varying the mass and initial velocity of the two carts. The first collision will be between a moving cart and a stationary cart of equal mass. The collisions in the next two trials will be between a moving cart and a stationary cart of unequal mass; the stationary cart will be the heavier cart in one trial and the lighter cart in the other. The last three trials will follow the same arrangement of masses, but both carts will now

be given a nonzero initial velocity.

To perform an elastic collision, we arrange the carts so that the ends with mag-nets face each other, as shown in the dia-gram to the left. Since the magnets in both carts have the same polarity, a repulsive magnetic force is produced, preventing the carts from making contact during the colli-

sion and resulting in a nearly perfect elastic collision*.

* * Magnetic field interactions are nearly, but not perfectly, conservative. However, the energy lost in such interac-tions is so small that, even if we wanted to account for it, we could not measure it with our equipment.

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The inelastic collisions we will perform will essentially be the same three distinct collisions we did for elastic colli-sions, only this time with the carts re-versed. To perform an inelastic collision, we simply arrange the carts so that the

ends with Velcro face each other, as shown in the diagram above. To reverse the orientation of the carts, simply detach the cart-string adapters from both carts and reattach them with the carts in the proper orientation. Although the carts can be reversed by passing them over each other, please do not do this! This puts a considerable amount of tension in the strings and can damage the pulleys. Once the carts have been reattached appropriately, check to make sure that the strings have retained the proper tension and remain free from obstruction.

When the carts collide, they will stick together and move as one object. Since the two carts will now be moving at the same rate, both rotary motion sensors should theoretically mea-sure the same velocity. In reality, however, each rotary motion sensor will measure a slightly dif-ferent velocity. We could obtain a more accurate value by taking the average of the two measure-ments, but we will naively assume the final velocities to be different and determine the final ve -locity for each cart independently (as we did for elastic collisions).

Data Collection using PASCO ’ s Capstone Software:

As velocity measurements are taken, they are dis-played graphically in Capstone. The graph below shows the velocities of the two cars for a collision between a moving car and a stationary cart. Notice that the rotary motion sen-sors do not directly record the velocity of the carts before or after the collision, they measure the velocity continuously. The moving cart is set in motion by giving it a light push, causing it to accelerate until it reaches some maximum ve-locity. Once the cart reaches this maximum velocity, its ve-locity will remain approximately constant until the colli-sion, decreasing only slightly due to friction from the track and pulleys. You may want to change the scale of the hori-

zontal axis so you can see the graph in more detail and determine the time of the collision more accurately. You can change the scale of an axis by placing the cursor over one of the numbers on the axis (the cursor will turn into arrows while placed over the axis) and dragging to the right (zoom in) or to the left (zoom out). You can also zoom in or out using the mouse wheel.

When the carts collide, their velocities will change abruptly, shown in the graph between and on the top graph and between and on the bottom graph. After a moment, the velocities of the carts will stabilize at some final velocity. We are interested in the velocity of each cart immedi-ately before and immediately after the collision, shown selected and labeled in the graph above. These velocities are what we will record as our initial and final velocities, respectively.

Notice that the velocity of each cart is fairly constant before and after the collision (i.e. the graph of their velocity is flat). The data points you should select for your initial velocities are therefore the last data points on the flat portion of each graph before the collision (i.e. the abrupt change in velocity). Likewise, the data points you should select for your final velocities should

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be the first data points on the flat portion of the graph after the collision. The RMSs are not nec-essarily perfectly synchronized and each cart may take a different amount of time to reach its fi -nal velocity, so do not worry if the initial velocities or the final velocities you select for the two carts do not occur at the same time (notice that the final velocities selected on the graph above do not occur at the same time). Just select the data points which appear to lie on the straight line im-mediately before and after the collision.

To determine the velocity of the carts at these times, first click on the graph and then click the “ ” button to use the selection tool. This will bring up a small box which you can move around to select the desired data point. As you move the box, an arrow will appear that points to the data point which is currently selected. Select a data point from just before the abrupt change in velocity and record this as the initial velocity for this cart. Then click the “ ” button again to select a data point right after the abrupt change in velocity on the same graph and record this as the final velocity for this cart. Once you have selected these two data points, click on the other graph and select the initial and final velocity for this cart in the same way.

Legend:

Elastic Collisions Inelastic Collisions

Trial Mass (Motion) → Mass (Motion) Trial Mass (Motion) → Mass (Motion)

1 Light (Moving) → Light (Stationary) 4 Light (Moving) → Light (Stationary)

2 Light (Moving) → Heavy (Stationary) 5 Light (Moving) → Heavy (Stationary)

3 Heavy (Moving) → Light (Stationary) 6 Heavy (Moving) → Light (Stationary)

Table I: Mass of Each Cart With/Without Adapter and Weights

Mass [g]

Cart Adapter WeightCart with Adapter

Cart withAdapter &

Weight

Blue 12.5

Red 12.5

1. Elastic Collisions:

NOTE: One of the biggest sources of error in this lab is not having a level track. If your track is not level, there will be an external force on your carts, which will cause them to accelerate. The next biggest source of error is having the string too tight. The string should be just loose enough to still turn the RMS pulley without slipping.

1. Assemble the experimental apparatus as described above for elastic collisions. Adjust the tension in the strings so the carts move smoothly. Make sure the strings are free from obstruc-tions (e.g. they do not rub on anything) as well.

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2. Using a digital scale, measure and record the mass of each cart in Table I. You may need to unscrew the cart-string adapter to remove the cart from the track. Measure and record the mass of the two weights, one with a blue dot and one with a red dot, in Table I as well and then fill in the rest of the table.

3. If it is not already running, open the Capstone file named “Elastic_Collisions.cap”.4. Click “Record” and gently push the cart closest to the Rotary Motion Sensors toward the

other cart. Make sure the carts do not hit the Rotary Motion Sensors. After the collision is over, click “Stop” to stop recording.

5. Determine the time at which the collision occurred using the graphical representation of the data, adjusting the scale as necessary as described above. Click inside the graph and then use the selection tool by clicking the “ ” button. Select a data point representing the initial veloc-ity on one of the graphs. Repeat this process until you have data for both carts, just before and just after the collision (four data points total). Record the data in Table II.

6. Repeat steps 4-5, but this time, add a weight (≈ 250g) to the stationary cart and push the cart hard enough so that the heavy cart moves with a reasonable (not extremely slow) velocity af-ter the collision. Record these measurements as Trial 2 in Table I.

7. Repeat steps 4-5, but this time, add a weight (≈ 250g) to the moving cart and push the cart gently enough so that the stationary cart moves with a reasonable (not extremely fast) velocity after the collision. Record these measurements as Trial 3 in Table I.

8. Now that you have completed all three collisions, use the data you have recorded to complete the other columns in Table II and Table III for Trials 1-3.

9. Once you have completed Table II and Table III and verified that the data is accurate enough and you do not need to repeat any trials, reverse the order of the carts so that the apparatus is now setup to perform inelastic collisions, as explained in the Experimental Apparatus section earlier in the handout.

2. Inelastic Collisions:

10. Repeat steps 4-8 with the carts now switched for inelastic collisions and record these as Trials 4-6 in Table I. When you finish, complete Table II and Table III.

QUESTION 1: What does it mean to say that some physical quantity (like energy or momentum) is conserved? How is conservation of energy different from conservation of momentum?

QUESTION 2: In this lab, we could define an internal force as one shared by the two carts and an external force as one shared by one cart (or the other) and its surroundings. Would we consider an internal force as one that would conserve momentum? Would we consider an external force as one that would conserve momentum? Explain.

QUESTION 3: What would be you most likely sources of error in your data? (Hint: Consider the external forces.)

QUESTION 4: From your experimental data, do you agree that elastic collisions conserve both momentum and energy? From your experimental data, do you agree that inelastic colli-sions conserve momentum but not energy? Explain.

QUESTION 5: If the change in energy () of a system due to a collision is negative, does this mean the system lost or gained energy?

EQUIPMENT:(1) Computer (2) PASCO 850 Interface

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(3) PASCO Dynamics Track(4) 2 – PASCars(5) 2 – PASCar Weights (Blue Dot & Red Dot)(6) 2 – Rotary Motion Sensors(7) 2 – Cart-String Adapters(8) 2 – IDS Mount Accessories

(9) 2 – IDS Track Pulley Brackets(10) Scales, AE Digital (600g/0.1g)(11) String(12) Excel File: "Momentum.xlsx"(13) Capstone Software File: “Elastic_Collisions.cap”

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Table II: Experimental Data

Cart Information Initial Final

Trial PASCarTotal mass

[g]Velocity[cm/s]

Momentum[g m/s]⋅

Kinetic Energy[mJ]

Velocity[cm/s]

Momentum[g m/s]⋅

Kinetic Energy[mJ]

Elastic Collisions

1

Blue

Red

Total

2

Blue

Red

Total

3

Blue

Red

Total

Inelastic Collisions

4

Blue

Red

Total

5

Blue

Red

Total

6

Blue

Red

Total

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Table III: Experimental Data AnalysisInitial Final Errors Conserved?

TrialTotal

Momen-tum

Total Ki-netic

Energy

TotalMomen-

tum

Total Ki-netic

EnergyMomentum Energy

MomentumConserved?

?

EnergyConserved?

?

Elastic

1

2

3

Inelastic

4

5

6

13