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MomentumMomentumForce is the Rate of Change of Momentum

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MomentumMomentum is a property of moving matter. Momentum describes the tendency of objects to keep going in the same direction with the same speed. Changes in momentum result from forces or create forces.

MomentumThe momentum of a ball depends on its mass and velocity. Ball B has more momentum than ball A.

Momentum and InertiaInertia is another property of mass that resists changes in velocity; however, inertia depends only on mass. Inertia is a scalar quantity.Momentum is a property of moving mass that resists changes in a moving objects velocity.Momentum is a vector quantity.

MomentumBall A is 1 kg moving 1m/sec, ball B is 1kg at 3 m/sec.A 1 N force is applied to deflect the motion of each ball. What happens? Does the force deflect both balls equally?Ball B deflects much less than ball A when the same force is applied because ball B had a greater initial momentum.

Kinetic Energy and MomentumKinetic energy and momentum are different quantities, even though both depend on mass and speed. Kinetic energy is a scalar quantity.Momentum is a vector, so it always depends on direction.Two balls with the same mass and speed have the same kinetic energy but opposite momentum.

Calculating MomentumThe momentum of a moving object is its mass multiplied by its velocity. That means momentum increases with both mass and velocity.

Velocity (m/sec)Mass (kg)Momentum (kg m/sec)

p = m v

You are asked for momentum.You are given masses and velocities.Use: p = m v Solve for car: p = (1,300 kg) (13.5 m/s) = 17,550 kg m/sSolve for cycle: p = (350 kg) (30 m/s) = 10,500 kg m/sThe car has more momentum even though it is going much slower.

Comparing momentumA car is traveling at a velocity of 13.5 m/sec (30 mph) north on a straight road. The mass of the car is 1,300 kg. A motorcycle passes the car at a speed of 30 m/sec (67 mph). The motorcycle (with rider) has a mass of 350 kg. Calculate and compare the momentum of the car and motorcycle.

Conservation of MomentumThe law of conservation of momentum states when a system of interacting objects is not influenced by outside forces (like friction), the total momentum of the system cannot change.If you throw a rock forward from a skateboard, you will move backward in response.

Conservation of Momentum

To see the relationship, consider two ballsconnected by a spring. The balls are motionless and therefore have no momentum.When you compress the spring, the third law says the balls exert equal forces(through the springs) in opposite directions on one another, -F1 = F2Remember from Newtons second law that the equal and opposite forces createopposite accelerations, which create opposite velocities. The accelerations are inversely proportional to the masses, so the velocities are also inverselyproportional to the masses. Heavy objects end up with less velocity and lightobjects with more velocity. The velocities caused by the original equal andopposite forces are exactly as predicted by the law of momentum conservation.

Collisions in One DimensionA collision occurs when two or more objects hit each other. During a collision, momentum is transferred from one object to another.Collisions can be elastic

or inelastic.

Collisions

Elastic collisionsTwo 0.165 kg billiard balls roll toward each other and collide head-on. Initially, the 5-ball has a velocity of 0.5 m/s. The 10-ball has an initial velocity of -0.7 m/s. The collision is elastic and the 10-ball rebounds with a velocity of 0.4 m/s, reversing its direction. What is the velocity of the 5-ball after the collision?

You are asked for 10-balls velocity after collision.You are given mass, initial velocities, 5-balls final velocity.Diagram the motion, use m1v1 + m2v2 = m1v3 + m2v4Solve for V3 : (0.165 kg)(0.5 m/s) + (0.165 kg) (-0.7 kg)=(0.165 kg) v3 + (0.165 kg) (0.4 m/s)V3 = -0.6 m/sElastic collisions

Inelastic collisionsA train car moving to the right at 10 m/s collides with a parked train car. They stick together and roll along the track.If the moving car has a mass of 8,000 kg and the parked car has a mass of 2,000 kg, what is their combined velocity after the collision?

You are asked for the final velocity.You are given masses, and initial velocity of moving train car.

Diagram the problem, use m1v1 + m2v2 = (m1v1 +m2v2) v3Solve for v3= (8,000 kg)(10 m/s) + (2,000 kg)(0 m/s)(8,000 + 2,000 kg)v3= 8 m/sThe train cars moving together to right at 8 m/s.

Inelastic collisions

Collisions in 2 and 3 DimensionsMost real-life collisions do not occur in one dimension. In a two or three-dimensional collision, objects move at angles to each other before or after they collide.In order to analyze two-dimensional collisions you need to look at each dimension separately. Momentum is conserved separately in the x and y directions.

Collisions in 2 and 3 Dimensions

Force is the Rate of Change of MomentumInvestigation Key Question:How are force and momentum related?

Force is the Rate of Change of MomentumMomentum changes when a net force is applied. The inverse is also true: If momentum changes, forces are created. If momentum changes quickly, large forces are involved.

Force and Momentum ChangeThe relationship between force and motion follows directly from Newton's second law.

Change in momentum(kg m/sec)Change in time (sec)Force (N)F = D p D t

You are asked for force exerted on rocket.You are given rate of fuel ejection and speed of rocketUse F = tSolve: = (100 kg) (-25,000 kg m/s) (1s) = - 25,000 NThe fuel exerts and equal and opposite force on rocket of +25,000 N.

Calculating forceStarting at rest, an 1,800 kg rocket takes off, ejecting 100 kg of fuel per second out of its nozzle at a speed of 2,500 m/sec. Calculate the force on the rocket from the change in momentum of the fuel.

ImpulseThe product of a force and the time the force acts is called the impulse. Impulse is a way to measure a change in momentum because it is not always possible to calculate force and time individually since collisions happen so fast.

Imagine a car hitting a wall and coming to rest. The force on the car due to the wall is large (big F ), but that force only acts for a small amount of time (little t ). Now imagine the same car moving at the same speed but this time hitting a giant haystack and coming to rest. The force on the car is much smaller now (little F ), but it acts for a much longer time (big t ). In each case the impulse involved is the same since the change in momentum of the car is the same. Any net force, no matter how small, can bring an object to rest if it has enough time. A pole vaulter can fall from a great height without getting hurt because the mat applies a smaller force over a longer period of time than the ground alone would.Stopping Time

F t = F t

Force and Momentum ChangeTo find the impulse, you rearrange the momentum form of the second law.

Change in momentum(kgm/sec)Impulse (Nsec)F D t = D p

Impulse can be expressed in kgm/sec (momentum units) or in Nsec.

Impulse - Momentum Example

A 1.3 kg ball is coming straight at a 75 kg soccer player at 13 m/s who kicks it in the exact opposite direction at 22 m/s with an average force of 1200 N. How long are his foot and the ball in contact?answer: Well use Fnet t = p. Since the ball changes direction, p = m v = m (vf - v0) = 1.3 [22 - (-13)] = (1.3 kg) (35 m/s) = 45.5 kg m /s. Thus, t = 45.5 / 1200 = 0.0379 s, which is just under 40 ms. During this contact time the ball compresses substantially and then decompresses. This happens too quickly for us to see, though. This compression occurs in many cases, such as hitting a baseball or golf ball.

Nearly all modern airplanes use jet propulsion to fly. Jet engines and rockets work because of conservation of linear momentum.A rocket engine uses the same principles as a jet, except that in space, there is no oxygen.Most rockets have to carry so much oxygen and fuel that the payload of people or satellites is usually less than 5 percent of the total mass of the rocket at launch.

Jet Engines

Sample ProblemAn electron (m= 9.1 x10-31 kg) moving at 2.18 x 106 m/s (as if it were in a Bohr orbit in the H atom).

Sample ProblemThere are 20 students in a school bus service with a mass of 2500 kg moving with a speed of 4.25 m/s. (a) What is the momentum of the bus and its passenger if the average mass of a student is 125 kg? (b) If 8 students get off from the bus, what is the momentum of the bus with its passengers if it continues to move with the same speed?

Sample ProblemHow fast must Superman fly to have the same momentum as a 2500 kg train moving at 4.5 m/s in the same direction? Assume that the mass of Superman is 120 kg.

Sample ProblemA system is made up of three bodies with their respective velocities: body A of mass 1.5 kg and moving East at 2.0 m/s; body B of mass 2.0 kg moving west at 3.0 m/s and body C of mass 5.2 kg moving west at 2.5 m/s. What is the momentum of the system?

Sample ProblemIf a 5 kg object experiences a 10N force for a 0.10 sec then, what is the momentum change of the object?

Sample ProblemA hockey player applies an average force of 80 N to a 0.25 kg hockey puck for a time of 0.10 seconds. Determine the impulse experienced by the hockey puck.

Sample ProblemWhat is the momentum of a system of two particles with these masses and velocities: 3.75 kg moving North at 5.6 m/s and 4.2 kg moving Northwest at 2.3 m/s?

Sample ProblemA 0.02 kg ball moving to the right at 0.25 m/s makes an elastic head-on collision with a 0.04 kg ball moving to the left at 0.15 m/s. After the collision, the smaller ball moves to the left at 0.16 m/s. What is the velocity of the 0.04 kg ball after collision?

Sample ProblemA 0.25 kg arrow with a velocity of 15 m/s to the east strikes and pierces the bulls eye of a 7.0 kg target. What is the final velocity of the combined mass