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Linear Momentum: Definition, Impulse, and Conservation

This chapter explores linear momentum, including its definition, impulse, and conservation. It also covers the center of mass and the motion of the center of mass. Additionally, it delves into various types of collisions, such as 1D and 2D, and elastic and inelastic collisions.

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Linear Momentum: Definition, Impulse, and Conservation

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  1. Chapter 7 Linear Momentum

  2. Linear Momentum • Definition of Momentum • Impulse • Conservation of Momentum • Center of Mass • Motion of the Center of Mass • Collisions (1d, 2d; elastic, inelastic) Chap07- Linear Momentum: Revised 3/3/2010

  3. F21 F12 m1 m2 Momentum From Newton’s Laws Consider two interacting bodies with m2 > m1: If we know the net force on each body then The velocity change for each mass will be different if the masses are different. Chap07- Linear Momentum: Revised 3/3/2010

  4. Momentum Rewrite the previous result for each body as: From the 3rd Law Combine the two results: Chap07- Linear Momentum: Revised 3/3/2010

  5. Define Momentum The quantity (mv) is called momentum (p). p = mv and is a vector. The unit of momentum is kg m/s; there is no derived unit for momentum. Chap07- Linear Momentum: Revised 3/3/2010

  6. Momentum From slide (4): The change in momentum of the two bodies is “equal and opposite”. Total momentum is conserved during the interaction; the momentum lost by one body is gained by the other. Δp1 +Δp2 = 0 Chap07- Linear Momentum: Revised 3/3/2010

  7. Impulse Definition of impulse: We use impulses to change the momentum of the system. One can also define an average impulse when the force is variable. Chap07- Linear Momentum: Revised 3/3/2010

  8. 30 N m1 or m2 Impulse Example Example (text conceptual question 7.2): A force of 30 N is applied for 5 sec to each of two bodies of different masses. Take m1 < m2 (a) Which mass has the greater momentum change? Since the same force is applied to each mass for the same interval, p is the same for both masses. Chap07- Linear Momentum: Revised 3/3/2010

  9. Example continued: (b) Which mass has the greatest velocity change? Since both masses have the same p, the smaller mass (mass 1) will have the larger change in velocity. (c) Which mass has the greatest acceleration? Since av the mass with the greater velocity change will have the greatest acceleration (mass 1). Chap07- Linear Momentum: Revised 3/3/2010

  10. Change in Momentum Example An object of mass 3.0 kg is allowed to fall from rest under the force of gravity for 3.4 seconds. Ignore air resistance. What is the change in momentum? Want p = mv. Chap07- Linear Momentum: Revised 3/3/2010

  11. Change in Momentum Example What average force is necessary to bring a 50.0-kg sled from rest to 3.0 m/s in a period of 20.0 seconds? Assume frictionless ice. The force will be in the direction of motion. Chap07- Linear Momentum: Revised 3/3/2010

  12. v1i v2i m1 m2 m1 m2 Conservation of Momentum m1>m2 A short time later the masses collide. What happens? Chap07- Linear Momentum: Revised 3/3/2010

  13. N2 N1 F12 F21 y w2 w1 x During the interaction: There is no net external force on either mass. Chap07- Linear Momentum: Revised 3/3/2010

  14. The forces F12 and F21 are internal forces. This means that: In other words, pi = pf. That is, momentum is conserved. This statement is valid during the interaction (collision) only. Chap07- Linear Momentum: Revised 3/3/2010

  15. Rifle Example A rifle has a mass of 4.5 kg and it fires a bullet of 10.0 grams at a muzzle speed of 820 m/s. What is the recoil speed of the rifle as the bullet leaves the barrel? As long as the rifle is horizontal, there will be no net external force acting on the rifle-bullet system and momentum will be conserved. Chap07- Linear Momentum: Revised 3/3/2010

  16. Center of Mass The center of mass (CM) is the point representing the mean (average) position of the matter in a body. This point need not be located within the body. Chap07- Linear Momentum: Revised 3/3/2010

  17. Center of Mass The center of mass (of a two body system) is found from: This is a “weighted” average of the positions of the particles that compose a body. (A larger mass is more important.) Chap07- Linear Momentum: Revised 3/3/2010

  18. Center of Mass In 3-dimensions write where The components of rcm are: Chap07- Linear Momentum: Revised 3/3/2010

  19. y x x A Center of Mass Example Particle A is at the origin and has a mass of 30.0 grams. Particle B has a mass of 10.0 grams. Where must particle B be located so that the center of mass (marked with a red x) is located at the point (2.0 cm, 5.0 cm)? Chap07- Linear Momentum: Revised 3/3/2010

  20. Center of Mass Example Chap07- Linear Momentum: Revised 3/3/2010

  21. y 2 x 1 3 Find the Center of Mass The positions of three particles are (4.0 m, 0.0 m), (2.0 m, 4.0 m), and (1.0 m, 2.0 m). The masses are 4.0 kg, 6.0 kg, and 3.0 kg respectively. What is the location of the center of mass? Chap07- Linear Momentum: Revised 3/3/2010

  22. Example continued: Chap07- Linear Momentum: Revised 3/3/2010

  23. Motion of the Center of Mass For an extended body, it can be shown that p = mvcm. From this it follows that Fext = macm. Chap07- Linear Momentum: Revised 3/3/2010

  24. Center of Mass Velocity Body A has a mass of 3.0 kg and vx = +14.0 m/s. Body B has a mass of 4.0 kg and has vy = 7.0 m/s. What is the velocity of the center of mass of the two bodies? Consider a body made up of many different masses each with a mass mi. The position of each mass is ri and the displacement of each mass is ri = vit. Chap07- Linear Momentum: Revised 3/3/2010

  25. Center of Mass Velocity For the center of mass: Solving for the velocity of the center of mass: Or in component form: Chap07- Linear Momentum: Revised 3/3/2010

  26. Center of Mass Velocity Applying the previous formulas to the example, Chap07- Linear Momentum: Revised 3/3/2010

  27. Collisions in One Dimension When there are no external forces present, the momentum of a system will remain unchanged. (pi = pf) If the kinetic energy before and after an interaction is the same, the “collision” is said to be perfectly elastic. If the kinetic energy changes, the collision is inelastic. Chap07- Linear Momentum: Revised 3/3/2010

  28. Collisions in One Dimension If, after a collision, the bodies remain stuck together, the loss of kinetic energy is a maximum, but not necessarily a 100% loss of kinetic energy. This type of collision is called perfectly inelastic. Chap07- Linear Momentum: Revised 3/3/2010

  29. Inelastic Collisions in One Dimension In a railroad freight yard, an empty freight car of mass m rolls along a straight level track at 1.0 m/s and collides with an initially stationary, fully loaded, boxcar of mass 4.0m. The two cars couple together upon collision. (a) What is the speed of the two cars after the collision? Chap07- Linear Momentum: Revised 3/3/2010

  30. Inelastic Collisions in One Dimension (b) Suppose instead that both cars are at rest after the collision. With what speed was the loaded boxcar moving before the collision if the empty one had v1i = 1.0 m/s. Chap07- Linear Momentum: Revised 3/3/2010

  31. One Dimension Projectile A projectile of 1.0 kg mass approaches a stationary body of 5.0 kg mass at 10.0 m/s and, after colliding, rebounds in the reverse direction along the same line with a speed of 5.0 m/s. What is the speed of the 5.0 kg mass after the collision? Chap07- Linear Momentum: Revised 3/3/2010

  32. Final Initial A A vAi B B Collision in Two Dimensions Body A of mass M has an original velocity of 6.0 m/s in the +x-direction toward a stationary body (B) of the same mass. After the collision, body A has vx = +1.0 m/s and vy = +2.0 m/s. What is the magnitude of body B’s velocity after the collision? Angle is 90o Chap07- Linear Momentum: Revised 3/3/2010

  33. Collision in Two Dimensions y momentum: x momentum: Solve for v2fy: Solve for v2fx: The mag. of v2 is Chap07- Linear Momentum: Revised 3/3/2010

  34. Summary • Definition of Momentum • Impulse • Center of Mass • Conservation of Momentum Chap07- Linear Momentum: Revised 3/3/2010

  35. Extra Slides Chap07- Linear Momentum: Revised 3/3/2010

  36. Conservation of Momentum Two objects of identical mass have a collision. Initially object 1 is traveling to the right with velocity v1 = v0. Initially object 2 is at rest v2 = 0. After the collision: Case 1: v1’ = 0 , v2’ = v0 (Elastic) Case 2: v1’ = ½ v0 , v2’ = ½ v0 (Inelastic) In both cases momentum is conserved. Case 1Case 2 Chap07- Linear Momentum: Revised 3/3/2010

  37. Conservation of Kinetic Energy? Case 1Case 2 KE After = KE Before KE After = ½ KE Before (Conserved) (Not Conserved) KE = Kinetic Energy = ½mvo2 Chap07- Linear Momentum: Revised 3/3/2010

  38. Where Does the KE Go? Case 1Case 2 KE After = KE Before KE After = ½ KE Before (Conserved) (Not Conserved) In Case 2 each object shares the Total KE equally. Therefore each object has KE = 25% of the original KE Before Chap07- Linear Momentum: Revised 3/3/2010

  39. 50% of the KE is Missing Each rectangle represents KE = ¼ KE Before Chap07- Linear Momentum: Revised 3/3/2010

  40. Turn Case 1 into Case 2 Each rectangle represents KE = ¼ KEBefore Take away 50% of KE. Now the total system KE is correct But object 2 has all the KE and object 1 has none Use one half of the 50% taken to speed up object 1. Now it has 25% of the initial KE Use the other half of the 50% taken to slow down object 2. Now it has only 25% of the the initial KE Now they share the KE equally and we see where the missing 50% was spent. Chap07- Linear Momentum: Revised 3/3/2010

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