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Impulse and Momentum: Definition, Theorems, and Examples

Learn about impulse and momentum in physics, including the impulse-momentum theorem and the principle of conservation of linear momentum. Includes examples and calculations.

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Impulse and Momentum: Definition, Theorems, and Examples

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  1. Ch7. Impulse and Momentum There are situations in which force acting on an object is not constant, but varies with time. Two new ideas: Impulse of the force and Linear momentum of an object.

  2. Impulse-Momentum Theorem

  3. Definition of Impulse The impulse J of a force is the product of the average force and the time interval during which the force acts: J = Impulse is a vector quantity and has the same direction as the average force. SI Unit of Impulse: newton.second (N.s)

  4. Definition of Linear Momentum: The linear momentum p of an object is the product of the object’s mass m and velocity v: p=mv Linear momentum is a vector quantity that points in the same direction as the velocity. SI Unit of Linear Momentum: kilogram.meter/second(kg.m/s)

  5. Impulse-Momentum Theorem When a net force acts on an object, the impulse of this force is equal to the change in momentum of the object: Final momentum Initial momentum Impulse Impulse=Change in momentum

  6. Example 1. A Well-Hit Ball • A baseball (m=0.14kg) has initial velocity of v0=-38m/s as it approaches a bat. The bat applies an average force that is much larger than the weight of the ball, and the ball departs from the bat with a final velocity of vf=+58m. • Determine the impulse applied to the ball by the bat. • Assuming time of contact is =1.6*10-3s, find the average force exerted on the ball by the bat.

  7. (a) = +13.4 kg.m/s (b)

  8. Example 2. A Rain Storm Rain comes straight down with velocity of v0=-15m/s and hits the roof of a car perpendicularly. Mass of rain per second that strikes the car roof is 0.06kg/s. Assuming the rain comes to rest upon striking the car (vf=0m/s), find the average force exerted by the raindrop.

  9. = -(0.06kg/s)(-15m/s)=0.9 N According to action-reaction law, the force exerted on the roof also has a magnitude of 0.9 N points downward: -0.9N

  10. Conceptual Example 3. Hailstones Versus Raindrops Suppose hail is falling. The hail comes straight down at a mass rate of m/ =0.06kg/s and an initial velocity of v0=15m/s and strikes the roof perpendicularly. Hailstones bounces off the roof. Would the force on the roof be smaller than, equal to, or greater than that in example 2? Greater.

  11. Check your understanding 1 Suppose you are standing on the edge of a dock and jump straight down. If you land on sand your stopping time is much shorter than if you land on water. Using the impulse-momentum theorem as a guide, determine which one is correct. A In bringing you to a halt, the sand exerts a greater impulse on you than does the water. B In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a greater average force. C In bringing you to a halt, the sand and the water exert the same impulse on you, but the sand exerts a smaller average force. B

  12. The Principle of Conservation of Linear Momentum

  13. Two types of forces act on the system: • Internal forces: Forces that the objects within the system exert on each other. • External forces: Forces exerted on the objects by agents external to the system. External force Internal force External force Internal force

  14. ( ) sum of average external forces sum of average internal forces = pf - p0 + internal forces cancel F12 = -F21 (Sum of average external forces) = pf - p0 If sum of external forces is zero (an isolated system) Then 0 = pf - p0 or pf = p0 m1vf1+m2vf2 = m1v01+m2v02 pf p0

  15. Principle of Conservation of Linear Momentum: The total linear momentum of an isolated system remains constant(is conserved). An isolated system is one for which the vector sum of the average external forces acting on the system is zero.

  16. Conceptual Example 4. Is the Total Momentum Conserved? • Two balls collide on the billiard table that is free of friction. • Is the total momentum of the two ball system the same before and after the collision? • Answer (a) for a system that contains only one ball. • The total momentum is conserved. (b)The total momentum of one ball system is not conserved.

  17. Example 5. Assembling a Freight Train Car 1 has a mass of m1=65*103kg and moves at a velocity of v01=+0.8m/s. Car 2 has a mass of m2=92*103kg and a velocity of v02=+1.3m/s. Neglecting friction, find the common velocity vf of the cars after they become coupled.

  18. (m1+m2) vf = m1v01 + m2v02 After collision Before collision =+1.1 m/s

  19. Example 6. Ice Skaters Starting from rest, two skaters push off against each other on smooth level ice (friction is negligible). One is a woman (m1=54kg), and one is a man(m2=88kg). The woman moves away with a velocity of vf1=2.5m/s. Find the recoil velocity vf2 of the man.

  20. F12 F21 internal forces system taken together 0 No external forces. isolated system conservation of momentum For the two skater system, what are the forces? In horizontal direction

  21. m1vf1 + m2vf2 = 0 after pushing before pushing It is important to realize that the total linear momentum may be conserved even when the kinetic energies of the individual parts of a system change.

  22. Check your understanding 2 A canoe with two people aboard is coasting with an initial momentum of 110kg.m/s. Then person 1 dives off the back of the canoe. During this time, the net average external force acting on the system is zero. The table lists four possibilities for the final momentum of person 1 and final momentum of person 2 plus the canoe, immediately after person 1 leaves the canoe. Which possibility is correct? a

  23. Collisions in One Dimension Elastic collision: One in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision. Inelastic collision: One in which the total kinetic energy of the system is not the same before and after the collision; if the objects stick together after colliding, the collision is said to be completely inelastic.

  24. Example 7. A Collision in One Dimension A ball of mass m1=0.25kg and velocity v01=5m/s collides head-on with a ball of mass m2=0.8kg that is initially at rest(v02=0m/s). No external forces act on the balls. If the collision in elastic, what are the velocities of the balls after the collision?

  25. Total momentum after collision Total momentum before collision Total kinetic energy before collision Total kinetic energy after collision (1) (2) (3)

  26. Substitute in (2)

  27. Substitute in (1)

  28. m1=0.25, m2=0.8 v01 =5 m/s, v02= 0

  29. Types of Collisions

  30. Example 8. A Ballistic Pendulum The ballistic pendulum consists of a block of wood(mass m2=2.5kg)suspended by a wire of negligible mass. A bullet(mass m1=0.01kg)is fired with a speed v01. After collision, the block has a speed vf and then swings to a maximum height of 0.65m above the initial position. Find the speed v01 of the bullet, assuming air resistance is negligible.

  31. Just before collision Just after collision m1+m2 vf Yes (no external forces ) Is conservation of energy valid? No (completely inelastic) Is conservation of momentum valid?

  32. Total momentum after collision Total momentum before collision

  33. hf=0.65 m vf Total mechanical energy at top of swing, all potential Total mechanical energy at bottom of swing, all kinetic Applying conservation of energy

  34. Check your understanding 3 Two balls collide in a one-dimensional, elastic collision. The two balls constitute a system, and the net external force acting on them is zero. The table shows four possible sets of values for the initial and final momenta of the two balls as well as their initial and final kinetic energies. Which one is correct?

  35. d

  36. Example 9. A Collision in Two Dimensions

  37. vf2=0.7 m/s Collisions in Two Dimensions x component y component

  38. x component P0x Pfx y component P0y Pfy

  39. Use momentum conservation to determine the magnitude and direction of the final velocity of ball 1 after the collision. x component Ball 1 after Ball 2 after Ball 2 before Ball 1 before

  40. y component Ball 1 after Ball 2 after Ball 1 before Ball 2 before

  41. Center of Mass

  42. Suppose m1=5kg, m2=12kg x1=2m, x2=6m

  43. During a time displacements of the particles, displacement of cm

  44. m1=0.25 kg, m2=0.8 kg v01=5 m/s, v02=0 m/s Before collision After collision

  45. Check your understanding 4 Water, dripping at a constant rate from a faucet, falls to the ground. At any instant, there are many drops in the air between the faucet and the ground. Where does the center of mass of the drops lie relative to the halfway point between the faucet and the ground: above it, below it, or exactly at the halfway point? (consider gravity) Above the halfway point

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