Lab 9 CONSERVATION OF LINEAR MOMENTUM In this experiment, you will try to verify the Law of Conservation of Linear Momentum using a collision between two cars on the air track. Here are the important concepts to understand: (a) THE LINEAR MOMENTUM P for a mass m is defined to be the product of the mass m and the velocity v, i.e., P = mv. P is a vector quantity and its MKS unit is kg m/sec. A cart moving to the right on the air track is considered to have a positive momentum. (b) AN ELASTIC COLLISION between two objects is defined as a collision where the mechanical energy of the system is conserved, i.e., the energy associated with the motion of the carts is not transformed into other forms of energy. In this experiment such a collision is produced by placing a spring between the two colliding carts. (c) AN INELASTIC COLLISION is produced when some of the mechanical energy of the carts is transformed into other forms of energy, e.g., heat, noise etc. An inelastic collision is produced by using two carts that stick together after the collision. Such a collision is known as a perfectly inelastic collision. Real collisions when the objects neither conserve their mechanical energy, nor stick to each other after the collision lie somewhere between the above two extreme cases. The Law of Conservation of Linear Momentum applies to ALL COLLISIONS. The purpose of this experiment is to demonstrate this fact. The conservation statement applies to the Total Momentum of the System P = P 1 + P 2 where: P 1 = momentum of the cart #1 and P 2 = momentum of cart #2 Formally the conservation law states that: In the absence of external forces, the total momentum of a system remains constant. (Total momentum remains the same before and after the collision.) The force between the two carts at the instant of collision is considered to be an internal force. More specifically, (a) AN INTERNAL FORCE is one that acts between two parts of the same system. Example: The force of collision between two carts. Here the system is considered to be the two carts taken together. As one cart pushes on the other the reaction force pushes back on the other cart. (b) AN EXTERNAL FORCE is a force between one or more parts of a given system and an external object. Example: Friction between carts (the system) and the air track (External). The conservation law is a direct consequence of Newton s 2nd and 3rd laws. Consider the collision between two carts. At the instant of collision, the cart #1 pushes cart #2 with a force F. F is not easily measured and is usually unknown to us. Cart #2, in return, pushes cart #1 back with an equal and opposite force, namely F. This is Newton s 3rd Law. (Each action is accompanied with an equal and opposite reaction.) Writing Newton s 2nd Law for the two cars, one arrives at: The lab instructions for this experiment were written by R. Warasila et al. of Suffolk County Community College. 1
Cart #1 Cart #2 Force = -F Force = F -F = m 1 a 1 = m 1 v 1 / t F = m 2 a 2 = m 2 v 2 / t -F = m 1 v 1 / t = (m 1 v 1 )/ t F = m 2 v 2 / t = (m 2 v 2 )/ t -F = P 1 / t F = P 2 / t Here, the masses are written as m 1 and m 2 and the velocities as v 1 and v 2, etc. Since m 1 and m 2 are constants, m 1 v 1 is written as (m 1 v 1 ). F is an unknown force. However, adding the final equations for the two carts results in: (m 1 v 1 ) (m 2 v 2 ) (m 1 v 1 + m 2 v 2 ) (P 1 + P 2 ) P tot -F + F = 0 = + = = = t t t t t Therefore, the quantity P tot = m 1 v 1 + m 2 v 2 is constant with time. This result is independent of the force F which is unknown. Procedure PART A: Elastic Collision Between Two Equal Masses: Choose two carts. Attach enough small masses to one to ensure equal masses for the two carts. Leave one at rest in the middle of the air track and set the other into motion. Measure the velocity and consequently the momentum for both carts BEFORE and AFTER the collision. The velocity measurements are to be done as in the previous experiments that utilize the air track. Compare the total momentum before and after the collision and give a % difference. Repeat the experiment with at least three different initial velocities for the moving cart. A spring at the end of the air track can be used to provide the initial velocities. Record m 1 = mass of cart #1, m 2 mass of cart #2, t for each cart and calculate v 1 and v 2, the velocities of the two carts and the momentum before and after the collisions. There are two photocells provided in this experiment to facilitate the measurement of the two velocities. Make sure you measure and use the L eff appropriate to each cart. PART B: Inelastic Collision Between Two Carts: Now choose two carts that stick together after the collision. Masses are not important. Equal or unequal masses are equally suitable but they must be measured. Repeat the experiment as before. Here the initial momentum comes from the moving cart and is given as m 1 v 1 (the stationary cart carries no momentum). The final momentum after the collision is given by: P after = m 1 v + m 2 v = (m 1 + m 2 )v Where v is the velocity of the two carts connected together. This part should also be done with three different initial velocities. 2
PART C: Elastic Collision Between Two Unequal Masses: In this trial send the smaller mass cart into the larger mass cart. The only difference between this part and Part A is that now the two masses are not equal. You will find that both carts will be in motion after the collision. You will therefore have to measure the final momentum from both carts. There are three velocities to measure (v 1 initially and v 1 and v 2 after the collision) and two timers to use. The same photocell is to be used for two measurements. Some quick reading of the timers and resetting is necessary here. This part of the experiment should also be done, using three different initial velocities. PART D: Elastic Collision Between Two Unequal Masses: In this trial send the larger mass cart into the smaller mass cart. Otherwise the experiment is the same as part C, so be ready to reset the timers as necessary. Summarize your results for the Law of Conservation of Linear Momentum in your report and verify that it applies to ALL COLLISIONS, perfectly elastic, perfectly inelastic and any collision in between these two extreme cases. PRELIMINARY QUESTIONS: Define momentum P and give its MKS unit here (in your words): Compute the momenta for each of the following particles and compute the total momentum of the three mass system: m 1 = 50 gm m 2 = 100 gm m 3 = 1 kg v 1 = 150 cm/s v 2 = 200 cm/s v 3 = 10 cm/s State the law of conservation of total linear momentum for a system COMPLETELY: For the system of a cart plus a small mass hanging in the air, as was done in the experiment on Newton s second law, draw all the forces on the system of two masses and identify which ones are internal and which ones may be considered to be external. Do this work below (or in a separate page to present in your report): 3
DATA AND PRESENTATION Use the data sheets for the 4 different collision setups. Elastic Collision of Two Equal Masses Inelastic Collision of Two Masses Elastic Collision of Two Unequal Masses - large cart hits small cart Elastic Collision of Two Unequal Masses - small cart hits large cart Conservation of the Total Momentum Error Table Trial P tot before P tot after % difference Elastic Collision Equal Masses Inelastic Collision Equal Masses Elastic Collisions Unequal Masses small into large Elastic Collisions Unequal Masses large into small Discuss how reasonable the % differences are in terms of the various sources of errors in the experiment. Include a conclusion regarding the conservation of momentum in elastic and inelastic collisions. Finally, answer the questions: 1) What was your initial expectation: P tot before > or < P tot after? WHY? 2) If P tot before > or < P tot after, make a list of the experimental effects that might contribute to this difference. Explain how each experimental effect can change the results (does it increase P tot after? Decreases it? Etc.) 3) What was/were the collision(s) with larger % difference? Suggest an explanation. 4
Elastic Collision of Two Carts of Equal Mass m 1 = m 2 = L eff = L eff = Before After Cart #1 Cart #2 Cart #1 Cart #2 t v 1 P 1 t v 2 P 2 P 1 + P 2 t v 1 P 1 t v 2 P 2 P 1 + P 2 0 0 0 5
Inelastic Collision of Two Carts of Equal Mass m 1 = m 2 = L eff = L eff = Before After Cart #1 Cart #2 Cart #1 Cart #2 t v 1 P 1 t v 2 P 2 P 1 + P 2 t v 1 P 1 t v 2 P 2 P 1 + P 2 0 0 0 6
Elastic Collision of Two Carts of Unequal Mass - light impacts heavy m 1 = m 2 = L eff = L eff = Before After Cart #1 Cart #2 Cart #1 Cart #2 t v 1 P 1 t v 2 P 2 P 1 + P 2 t v 1 P 1 t v 2 P 2 P 1 + P 2 7
Elastic Collision of Two Carts of Unequal Mass - heavy impacts light m 1 = m 2 = L eff = L eff = Before After Cart #1 Cart #2 Cart #1 Cart #2 t v 1 P 1 t v 2 P 2 P 1 + P 2 t v 1 P 1 t v 2 P 2 P 1 + P 2 8