Momentum

Momentum: Video Analysis Background When we talk about momentum , it can be first helpful to define another quantity: impulse . Impulse is a vector (direction matters) represented by There are two different ways to define impulse: The first is that impulse equals the change in momentum, or . The second is using the net force on the object and time, or . Combining these, we get and if you divide both sides by time, This means that if there is no net force on the body or system of bodies ( ⃗ F net = 0 ), then Δ ⃗ p Δt = 0 . Time cannot be zero, so there must be no change of momentum within the system with time. This is known as conservation of momentum . Let’s take conservation of momentum one step farther. Since the change in momentum has to be zero, or Since we are talking about the whole system, remember that the final momentum is the sum of the final momentum of the parts, and the initial momentum is the sum of the initial momentum of the parts. In today’s lab, you’ll be examining collisions. There are two types of collisions that will be examined. Perfectly inelastic collisions are collisions that hit, stick together, and move with a constant velocity after the collision. Elastic collisions are those where the objects do not stick together, but rather move separately from each other after the collision. (Elastic collisions also conserve kinetic energy.) Goal: The purpose of this experiment is to use LoggerPro to measure the momentum in various collisions and determine whether conservation of momentum applies. Case I - Perfectly Inelastic Collision with Equal Masses  Open LoggerPro  Insert Movie  Logger Files (on Desktop)  Collision Videos  Colliding_Carts_37-mmi.mov. Go to Page  Auto Arrange to put your movie

in the top left-hand corner. Set your scale and note the masses of the two carts from the first slide.  Plot the location of the left cart by using the Add Point button. Click on a certain part of the cart (front, middle, back…) and the movie will advance to the next frame. Be consistent, plot the same part of the cart for every frame, and continue plotting the points until the end of the movie.  Rewind the movie to the beginning. Press the Set Active Point button and select Add Point Series and plot the location of the cart on the right. You will notice that the dot is a different color. Start at the first slide and continue plotting the location, even if the cart is not moving.  Use LoggerPro to determine the velocity of each cart by taking the slope of the position v. time graph before and after the collision. Record these values in the table below.  Determine the momentum of each cart by going to Data  New Calculated Column. Name your new quantity “cart 1 p” or something similar. You will create an equation with constants and known variables. Note: to use a variable from your data, choose it from the Variables column. (For example, “ X Velocity ”) can be used to calculate the momentum in the x-direction. o Plot a “cart 1 p” v. time graph. Shade the part of the data that is a flat line before the collision and find the mean value by going to Analyze  Statistics. Do the same for after the collision. Record these values in the table below. Mass (kg) v i (m/s) p i (Ns) Σ p i (Ns) Cart left 0.524 0.5743 0.3009 0.3009 Cart right 0.524 0 0 Mass (kg) v f (m/s) p f (Ns) p Σ f (Ns) p Δ (Ns) Two carts together 1.048 0.283 0.296 0.296 -0.0049 3 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

Case III - Perfectly Elastic Collisions with Equal Masses  Repeat the experiment above, this time using the Colliding_Carts_41-mme.mov file. Mass (kg) v i (m/s) p i (Ns) Σ p i (Ns) Cart left 0.524 0.4860 0.2546 0.2546 Cart right 0.524 0 0 Mass (kg) v f (m/s) p f (Ns) p Σ f (Ns) p Δ (Ns) Cart left 0.524 0 0 0.2496 -0.005 Cart right 0.524 0.4762 0.2496 Case IV - Perfectly Elastic Collisions with Unequal Masses  Repeat the experiment above, this time using the Colliding_Carts_45-mMe.mov file. Mass (kg) v i (m/s) p i (Ns) Σ p i (Ns) Cart left 0.524 -0.4832 -0.2531 -0.2531 Cart right 1.048 0 0 Mass (kg) v f (m/s) p f (Ns) p Σ f (Ns) p Δ (Ns) Cart left 0.524 -0.1386 -0.0726 0.2553 0.002 Cart right 1.048 0.3129 0.3279  Repeat the experiment above, this time using the Colliding_Carts_31-Mme.mov file. Mass (kg) v i (m/s) p i (Ns) Σ p i (Ns) Cart left 1.048 0.5099 0.5343 0.5343 Cart right 0.524 0 0 Mass (kg) v f (m/s) p f (Ns) p Σ f (Ns) p Δ (Ns) Cart left 1.048 0.6968 0.7302 0.0610 -0.4733 5 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

Cart right 0.524 0.0852 0.0446 Examining Momentum Conservation If a system had no net outside force on it, one would expect conservation of momentum and Δp=0, always. Since we are measuring momentums of bodies, before and after collisions, there can be errors in measurement. Likewise, if there are net outside forces, then Δp≠0. A measurement of error would be to compare Δp with ∑ | p i | . Since the cart on the right was always initially at rest, we can compare Δp with p left cart, initial .  Using the data you’ve collected, complete the table below to consider whether momentum was conserved in the collisions above. p left cart, initial (Ns) Δp (Ns) Δ p p ¿ ,initial × 100% Case 1 (mmi) 0.3009 -0.0049 -1.628 Case 2 (mMi) 0.286 -0.01 -3.496 Case 3 (Mmi) 0.4100 -0.02 -4.878 Case 4 (mme) 0.2546 -0.005 -1.96 Case 5 (mMe) -0.2531 0.002 -0.7902 Case 6 (Mme) 0.5343 -0.4733 -88.587 6 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

4. Two bodies on a frictionless surface come together, collide, and separate. Fill out the rest of the table below. Mass (kg) v i (m/s) p i (Ns) Σ p i (Ns) Cart left 0.600 +5.50 3.3 0.3 Cart right 0.400 -7.50 -3 Mass (kg) v f (m/s) p f (Ns) p Σ f (Ns) p Δ (Ns) Cart left 0.600 -1.50 -0.9 0.3 0.00 Cart right 0.400 3 1.3 8 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

Momentum: Collision Carts Background When we talk about momentum , it can be first helpful to define another quantity: impulse . Impulse is a vector (direction matters) represented by There are two different ways to define impulse: The first is that impulse equals that change in momentum, or . The second is using the net force on the object and time, or . Combining these, we get and if you divide both sides by time, This means that if there is no net force of the body or system of bodies ( ⃗ F net = 0 ), then Δ ⃗ p Δt = 0 . Time cannot be zero, so there must be no change of momentum within the system with time. This is known as conservation of momentum . Let’s take conservation of momentum one step farther. Since the change in momentum has to be zero, or Since we are talking about the whole system, remember that the final momentum is the sum of the final momentum of the parts, and the initial momentum is the sum of the initial momentum of the parts. In today’s lab, you’ll be creating collisions. There are two types of collisions that will be examined. Perfectly inelastic collisions are collisions that hit, stick together, and move with a constant velocity after the collision. Elastic collisions are those where the objects do not stick together, but rather move separately from each other after the collision. In perfectly elastic collisions, the total kinetic energy of the objects is also conserved. Goal: The purpose of this experiment is to measure the momentum in various collisions and determine whether conservation of momentum applies. 9 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

Part II - Elastic Collisions Now flip the carts around so that the magnetic sides will be near each other during the collision. Ensure that the collision is truly elastic by pushing the carts lightly enough that they do not actually hit (the magnetic repulsion is sufficient for redirecting the carts).  Repeat the above experiments with a range of conditions. Use the table at the end of this report to record your raw data and to carry out the calculations. Summary Question Describe what you learned about conservation of moment. 11 Phys 103 - Principles of Physics I Labs Momentum: Collision Carts

Part I Data - Inelastic Collisions Part II Data - Elastic Collisions Phys 103 - Principles of Physics I Labs Trial #1 0.265 0.274 0.34 -0.38 0 0 -0.143 0 -0.01436 Trial #2 0.2523 0.274 0.53 -0.37 0.18 0.18 0.18058 0.14508 -0.0355 Trial #3 0.2523 0.527 0.32 -0.32 0 0 -0.0869 0 0.0896 Trial #4 0.264 0.527 0.29 -0.31 -0.07 -0.07 -0.0868 -0.0553 0.0315 Trial #1 0.264 0.274 0.35 -0.28 -0.28 0.29 0.0157 0.00554 -0.010 Trial #2 0.2523 0.274 0.32 -0.25 0 0.43 0.09886 0.11782 0.01896 Trial #3 0.2523 0.527 0.39 -0.26 -0.24 0.33 -0.0386 0.1133 0.1519 Trial #4 0.264 0.527 0.29 -0.33 -0.55 0 -0.09735 -0.1452 -0.4785