Chapter 6: Momentum 12.1 Momentum 12.2 Force is the Rate of Change of Momentum 12.3 Angular Momentum 1

Chapter 12 Objectives Calculate the linear momentum of a moving object given the mass and velocity. Describe the relationship between linear momentum and force. Solve a one-dimensional elastic collision problem using momentum conservation. Describe the properties of angular momentum in a system—for instance, a bicycle. Calculate the angular momentum of a rotating object with a simple shape. 2

Chapter Vocabulary angular momentum collision law of conservation of elastic collision gyroscope impulse inelastic collision linear momentum

Inv 12.1 Momentum Investigation Key Question: What are some useful properties of momentum? 5

12.1 Momentum Momentum is a property of moving matter. Momentum describes the tendency of objects to keep going Net forces change momentum.

12.1 Momentum The momentum depends on mass and velocity. Ball B has more momentum than ball A.

12.1 Momentum If both ball A and B were pushed with the same force, what can we determine about their difference? Ball B deflects much less than ball A when the same force is applied because ball B had a greater initial momentum.

12.1 Kinetic Energy and Momentum Kinetic energy and momentum are different, 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 can have the same kinetic energy but opposite momentum.

12.1 Calculating Momentum p = m v Momentum (kg m/sec) Velocity (m/sec) Mass (kg)

Comparing momentum You are asked for momentum. Calculate and compare the momentum of the car and motorcycle. 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/s Solve for cycle: p = (350 kg) (30 m/s) = 10,500 kg m/s The car has more momentum even though it is going much slower.

12.1 Conservation of Momentum The law of conservation of momentum states that without an outside force, the total momentum of the system is constant. If you throw a rock forward from a skateboard, you will move backward in response.

12.1 Conservation of Momentum To see the relationship, consider two balls connected 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 = F2 Remember from Newton’s second law that the equal and opposite forces create opposite accelerations, which create opposite velocities. The accelerations are inversely proportional to the masses, so the velocities are also inversely proportional to the masses. Heavy objects end up with less velocity and light objects with more velocity. The velocities caused by the original equal and opposite forces are exactly as predicted by the law of momentum conservation.

12.1 Collisions in One Dimension A collision is when two or more objects hit each other. During a collision, momentum is transferred Collisions can be elastic or inelastic.

12.1 Collisions

Inelastic collisions 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) v3 Solve for v3= (8,000 kg)(10 m/s) + (2,000 kg)(0 m/s) (8,000 + 2,000 kg) v3= 8 m/s

The Archer An archer at rest on frictionless ice fires a 0.5-kg arrow horizontally at 50.0 m/s. The combined mass of the archer and bow is 60.0 kg. With what velocity does the archer move across the ice after firing the arrow? September 16, 2018

12.1 Collisions in 2 and 3 Dimensions Most real-life collisions do not occur in one dimension. 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.

12.1 Collisions in 2 and 3 Dimensions

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

12.2 Force is the Rate of Change of Momentum Momentum 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.

12.2 Force and Momentum Change The relationship between force and motion follows directly from Newton's second law. Force (N) F = D p D t Change in momentum (kg m/sec) Change in time (sec)

Calculating force You are asked for force exerted on rocket. Starting at rest, an 1,800 kg rocket takes off, ejecting 100 kg of fuel over a second at a speed of 2,500 m/sec. Calculate the force on the rocket from the change in momentum of the fuel. You are asked for force exerted on rocket. You are given rate of fuel ejection and speed of rocket Solve: Δ = (100 kg) (-2,500 m/s) = -250,000 kg m/s Use F = Δ ÷Δt Solve: ΔF = (100 kg) (-250,000 kg m/s) ÷(1s)= - 25,000 N The fuel exerts and equal and opposite force on rocket of +25,000 N.

12.2 Impulse Impulse measures a change in momentum because it is not always possible to calculate force and time individually Collisions happen so fast!

12.2 Force and Momentum Change To find the impulse, you rearrange the momentum form of the second law. Impulse (N•sec) F D t = D p Change in momentum (kg•m/sec) Impulse can be expressed in kg•m/sec (momentum units) or in N•sec.

Show from 5-6 min

12.2 Impulse What is the change of momentum between these two balls? Impulse = change in v * time aka change in momentum

12.2 Impulse What is the change of momentum between these two balls? Rubber ball change in velocity is 4. Clay one the change is 2. So the rubber ball had twice as much impulse

12.2 Impulse So things that bounce have a great impulse, so they feel a greater force!

12.2 Impulse You are given a choice at a carnival game of what ball to throw at stacked milk jugs. Sandbag which stops when it hits A base ball which goes through And a rubber ball which bounces

Take a vote 12.2 Impulse Which one do you choose? Sandbag which stops when it hits A base ball which goes through And a rubber ball which bounces Take a vote

12.2 Impulse You choose the rubber ball because the change in momentum is the most so the impulse is bigger. This means the force the ball feels is more. And with equal and opposite the jugs feel more force too!

12.2 Impulse But which ball do they let you use in this game?

12.2 Impulse This is why carnivals are evil