# Energy and Momentum Lessons

## Middle School Physical Science - Energy and Velocity

Next Generation Science Standards covered in unit:

MS.PS.2: Plan an investigation to provide evidence that the change in an objectâ€™s motion depends on the sum of the forces on the object and the mass of the object.

MS.PS3.1: Construct and interpret graphical displays of data to describe the relationships of kinetic energy to the mass of an object and to the speed of an object.

MS.PS3.5: Construct, use, and present arguments to support the claim that when the kinetic energy of an object changes, energy is transferred to or from the object.

ETS1.A: Defining and Delimiting an Engineering Problem

ETS1.B: Developing Possible Solutions

## A Quantitative PocketLab Study of Momentum, Impulse, and Force in the Collision of Two Carts

You don't need an expensive air track to do a quantitative study of momentum, impulse, and force involved in the collision of two carts.  You can get very good results by the use of two PocketLabs, two iPhones, and a pair of carts from the PocketLab Maker Kit.

As shown in the picture below, each cart has a PocketLab mounted on one of its wheels, so that the z-axis is the axis of interest when the carts are moving.  You can only connect one PocketLab to an iPhone at time, thus the need for two iPhones, each running the VelocityLab app.  Since we are interested only in velocity, it really makes no difference where on the track the zero position is for each of the carts.  The track, made from a pair of rails with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This helps to keep the carts on "the straight and narrow" as they move. Styrofoam bumpers at the ends of the tracks keep the carts from falling off of the table.  The collision is buffered by an 8" cable tie that is stuck (using thick double stick mounting tape) to one of the wood blocks that have been taped to each cart.  Cart B has a larger extra block of wood taped to it to give it higher mass.  The carts are given initial pushes in any direction desired.  The VelocityLab apps will display movement to the right as positive velocity and movement to the left as negative velocity.  The velocity graphs will likely not be synced in time, as it is quite difficult to start the VelocityLab apps on both iPhones at exactly the same time. But this is OK, as we only need to know the initial and final velocities as well as the difference between end and start times for the collisions.  Knowing these things will allow us to compute impulse and momentum changes as well as the average force during the collision.

The Excel table in the figure below provides all of the detailed calculations for four different scenarios for cart A and cart B.  All of the items in blue are raw data that are quickly obtained from a balance and the VelocityLab apps.  The impulse time should be the same for both carts, so the white column shows the average impulse time which is used in computing the force, shown in the red columns.  The green columns show the initial and final system (cart A and cart B combined) momenta.  The yellow columns show the change in momenta for each of the carts.  We note that for each of the four trials:

(1) The initial and final system momenta are reasonably close to being equal, as we would expect from the law of conservation of momentum.
(2) The change in the momenta for each of the carts are similar in magnitude but opposite in sign.
(3) The average force acting on each of the carts is also similar in magnitude but opposite in sign, as we would expect from Newton's Third Law of Motion.

As an example, the velocity vs. time graphs below show the details associated with trial 2 in the above table.  From the graphs and the red markup shown on the graphs, it should be clear how initial and final velocities as well as impulse times were obtained for each of the two carts.

As can be seen from the equations in the table showing the four trial results, the calculations are straight-forward but a bit tedious.  Therefore a copy of the Excel spreadsheet is provided as an attachment.  Your students can simply do scenarios of their own choosing and enter raw data into the blue columns.  Excel will then do all the calculations for the remaining columns.

## Conservation of Momentum When Two Carts "Explode"

Do you have two carts from the PocketLab Maker Kits?  Do you have two PocketLabs?  You probably have at least two students in your physics class with iPhones.  Do they have the VelocityLap app installed on their iPhones?  Then you have the major components for your students to investigate conservation of momentum when two carts on a track "explode".

As shown in the picture below, each cart has a PocketLab mounted on one of its wheels, so that the z-axis is the axis of interest when the carts are moving.  You can only connect one PocketLab to an iPhone at time, thus the need for two iPhones, each running the VelocityLab app.  The carts are shown in their zero positions at the center of a track on a table.

The track, made from a pair of rails with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This helps to keep the carts on "the straight and narrow" as they move.  The "explosion" is provided by a cable tie that is stuck (using thick double stick mounting tape) to one of the wood blocks that have been taped to each cart.  The carts are pushed together, compressing the cable tie, and then released. The stored elastic potential energy of the cable tie provides the energy for the
"explosion", causing the two carts to move off in opposite directions.

The photo below shows a closeup of the cable ties.  As can be seen from the NSTA ruler, they are about 8" long before looping the tie.  The looped cable tie that was used in the momentum experiments is also shown in the photo.

The total momentum of the system is zero before the explosion.  If momentum is conserved, we would expect it to be zero after the collision as well.  The figure below shows the velocity graphs obtained from data from the VelocityLab app for both carts A (on the left) and B (on the right).  These two graphs are really the only ones needed as they provide the velocity of the carts after the explosion.  These two graphs will likely not be synced in time as it is quite difficult to start the VelocityLab apps on both iPhones at exactly the same time. But this is OK, as we only need to know the final velocities and not the time at which they occurred.

The mass of cart A is 0.159 kg.  The mass of cart B, which has an extra wood block taped to the cart, is 0.223 kg.  The most important data on the charts have been highlighted with a star.  The graph shows that the velocity of cart A just after the explosion is -0.323 m/s.  Therefore, its momentum  = mv = -0.051 kg-m/s.  Similarly, the graph shows that the velocity of cart B just after the explosion is 0.244 m/s, and has a momentum = pv = 0.054 kg-m/s.  The sum of the final momenta is 0.003 kg-m/sec, very close to the initial zero momentum of the system before the explosion.  We have good evidence that the law of conservation of momentum is conserved even in explosions!  In explosions the vector sum of the exploded pieces is the same as the initial momentum of the system.  It is certainly worth while to ask students to think of real world examples of such explosions.

Below is a video of a typical run of the experiment:

### A Classic Conservation of Momentum Experiment with PocketLab

A well-known conservation of momentum experiment that has been around for many years involves dropping a brick onto a horizontally moving cart.  With PocketLab and the VelocityLab app, your students can perform this experiment easily with the cart from the PocketLab Maker Kit and a small block of wood.  The snapshot below shows the setup with the author about to drop the block of wood onto the cart coming from the left.  A pair of rails, with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This is optional but does help to keep the cart on "the straight and narrow".

The video below shows what a typical run of the experiment looks like.  The cart is given an initial push at the far left end of the track, receives the wood block near the middle of the track, and then the combined cart and wood block hit the styrofome bumper at the right end of the track.

The figure below shows position, velocity, and accelertion vs. time graphs that were constructed in Excel from data obtained from a .csv file created by the VelocityLab app.  The graphs are marked up in red, noting how the motion of the cart applies to the graphical interpretation of the data.

The two most important pieces of information needed to verify the law of conservation of momentum are identified with stars in the velocity graph: (1) the speed of the cart just before receiving the dropped wood block, and (2) the speed of the combined cart and wood block just after the receiving the wood block.  (Note that the mass of cart is 0.144 kg and the mass of the wood block is 0.096 kg.)

With all of the above information, it is quite easy to compute the momentum of the system right before the cart receives the wood block [p = mv = 0.144 kg x 0.841 m/s = 0.121 kg-m/s], and right after the wood block is dropped [p = mv = (0.144 kg + 0.096 kg) x 0.524 m/s = 0.126 kg-m/s].  The momentum before and after agree within about 4%, providing good verification of the law of conservation of momentum.

(A VelocityLab.csv file for a typical run of this experiment is attached for those who may be interested in viewing it.)

### A Momentum Conservation Experiment for an Inelastic Collision Between Two Carts

Do you have two PocketLab Maker Kit carts, and do you have the free VelocityLab app?  Then you are all set to do some experiments in conservation of momentum with PocketLab!  This lab discusses how to setup and perform an inelastic collision in which one cart (A) is moving toward another cart (B) that is at rest.  When cart A hits cart B, they stick and move off together.  The photo below shows the two carts shortly before the collision would occur.  PocketLab is mounted on a front wheel of cart A.  Small pieces of wood are stuck to the carts and protrude further than the wheels.  Some thick double-stick mounting tape is attached to the ends of the wood that overhang the wheels.  When the carts collide, the tapes stick together, providing an inelastic collision.  In such a collision some of the kinetic energy of the carts is converted into other forms of energy, including heat and sound.

The figure below shows the setup used by the author to study conservation of momentum.  A pair of rails, with inside separation just a little larger than the axle of the carts, was constructed with thin balsa wood sticks.  This is optional but does help to keep the carts on "the straight and narrow".  Cart A is given an initial push at the far left end of the track, while cart B is at rest near the middle of the track.  After sticking together, the carts hit the white Styrofoam bumper at the right end, recoil some and then stop.

The figure below shows position, velocity, and acceleration vs. time graphs that were constructed from data obtained from a .csv file created by the VelocityLab app.  The graphs are marked up in red, noting how the motion of the carts applies to the graphical interpretation of the data.

The two most important pieces of information needed to verify the law of conservation of momentum are identified with stars in the velocity graph: (1) the speed of cart A just before colliding and sticking to cart B, and (2) the speed of the combined carts just after sticking together.  (Note that the mass of cart A is 0.159 kg and the mass of cart B is 0.102 kg.)

With all of the above information, it is quite easy to compute the momentum of the system right before the collision [p = mv = 0.159 kg x 0.802 m/s = 0.128 kg-m/s], and right after the collision [p = mv = (0.159 kg + 0.102 kg) x 0.453 m/s = 0.118 kg-m/s].  The momnetum before and after agree within about 8%.  Calculations of the kinetic energy of translation reveal that about half of this kinetic energy is lost:  0.051 J just before the collision, and 0.027 J just after the collision.

The VelocityLab.csv file has been attached for those that may be interested in it.  The video below is a combined video and data, showing graphs of position, velocity, and acceleration in sync with the motion of the carts.

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