Momentum

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Uses the Law of Conservation of Momentum and describes momentum, impulse, elastic and inelastic collisions as well as explosions.
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Momentum

  1. 1. Copyright Sautter 2015
  2. 2. Newton’s Second Law of Motion Acceleration = velocity / time Combining the two equations Rearranging the equation Impulse Momentum 2
  3. 3. Impulse & Momentum • As seen on the previous slide, momentum and impulse equations are derived from Newton’s Second Law of Motion (F = ma). Impulse is defined as force times time interval during which the force is applied and momentum is mass times the resulting velocity change. • The symbol p is often used to represent momentum, therefore p = mv. • The question maybe however, “why develop a momentum equation when it is merely a restatement of F = ma which we already know ?” • The answer is that no conservation principles can be applied to forces. There is no such thing as “conservation of force”. However, Conservation of Momentum is a fundamental law which can be applied to a vast array of physics problems. Therefore, manipulating Newton’s Second Law into the impulse – momentum format helps us to implement a basic principle of physics ! 3
  4. 4. A collision is a momentum exchange. In all collisions the total momentum before a collision equals the total momentum after the collision. Momentum is merely redistributed among the colliding objects. Momentum = Mass x Velocity p = m x v Σ Momentum before = Σ Momentum after collision collision 4
  5. 5. Momentum & Kinetic Energy • Recall the following facts about kinetic energy: • Kinetic energy is the energy of motion. In order to possess kinetic energy an object must be moving. • As the speed (velocity) of an object increases its kinetic energy increases. The kinetic energy content of a body is also related to its mass. Most massive objects at the same speed contain most kinetic energy. • Since the object is in motion, the work content is called kinetic energy and therefore: K.E. = ½ m v2 5
  6. 6. Types of Collisions (Momentum Transfers) • Collisions occur in three basic forms: • (1) Elastic collisions – no kinetic energy is lost during the collision. The sum of the kinetic energies of the objects before collision equals the sum of the kinetic energies after collision. • It is important to realize that kinetic energy is conserved in these types of collisions. Total energy is conserved in all collisions. (The Law of Conservation of Energy requires it!) • The only collisions which are perfectly elastic are those between atoms and molecules. In the macroscopic world some collisions are close to perfectly elastic but when examined closely, they do lose some kinetic energy. 6
  7. 7. Types of Collisions (Momentum Transfers) • (2) Inelastic collisions – the objects stick together after collision and remain as a combined unit. Kinetic energy is not conserved ! • (3) Partially elastic collisions – kinetic energy is not conserved but the colliding objects do not remain stuck together after collision. Most collisions are of this type. In these collision some of the energy is converted to heat energy or used to deform the colliding objects. That which remains is retained as kinetic energy. 7
  8. 8. Types of Collisions (Momentum Transfers • Example collisions: • Elastic collisions – pool balls colliding, a golf ball struck with a club, a hammer striking an anvil. Remember, these collisions are not perfectly elastic but they are close! • Inelastic collisions – an arrow shot at a pumpkin and remaining embedded, a bullet shot into a piece of wood and not fully penetrating, two railroad cars colliding and coupling together. • Partially elastic collisions - two cars colliding and then separating, a softball struck by a bat, a tennis ball hit with a racket. 8
  9. 9. M1U1 + M2U2 = M1V1 + M2V2 CLICK HERE Σ K.E. before collision = Σ K.E. after collision e = 1.0 9
  10. 10. M1U1 + M2U2 = (M1 + M2) V CLICK HERE Σ K.E. before collision = Σ K.E. after collision/ e = 0 10
  11. 11. Measuring Collision Elasticity • The elastic nature of a collision can be measured by comparing the K.E. content of the system before and after collision. The closer they are to being equal, the more elastic the collision. • Another way to measure elasticity is to use the Coefficient of Restitution. The symbol for this value is e. If e is equal to 1.0 the collision is perfectly elastic. If e is equal to zero the collision is inelastic. If e lies in between 0 and 1.0 the collision is partially elastic. • The closer e is to 1.0 the more elastic the collision, the closer e is to 0 the more inelastic. 11
  12. 12. Ball 1 Ball 2 Velocity of ball 1 Before collision u1 Velocity of ball 2 Before collision u2 Ball 1 Ball 2 Point of collision Velocity of ball 1 after collision V1 Velocity of ball 2 after collision V2 12
  13. 13. Momentum – A Vector Quantity • Momentum is a vector quantity (direction counts). In one dimensional collisions motion lies exclusively along the x plane or the y plane. • One Dimensional Collisions - The direction of the momentum vectors are identified by the usual conventions of plus on horizontal plane (x axis) is to the right and minus is to the left. In the vertical plane (y axis) , plus is up and minus is down. • Two Dimensional Collisions – The momentum vectors can be resolved into x – y components and combined using vector addition methods to obtain the momentum sums before and after collision. • In two dimensional collisions the sums of the vertical momentum components must be equal before and after collision just as the sums of horizontal momentum components must be equal before and after collision. 13
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