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T HE I MMACULATE R ECEPTION

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MOMENTUM AP Physics C: Mechanics

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W HAT IS M OMENTUM ? What is its definition? How do we calculate it? When do we use this term? Why was this word invented? What do we already know about it? What do we want to know about it?

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W HAT IS M OMENTUM ? What is its definition? Momentum: the product of an objects mass and its velocity Momentum: mass in motion Momentum: quantity of motion -Newton Momentum: It is a vector! Momentum: is sometimes called linear momentum

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W HAT IS M OMENTUM ? How do we calculate it? What are its units? If object is moving in arbitrary direction:

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W HAT DO WE KNOW ABOUT MOMENTUM ?

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W HAT IS M OMENTUM ? Why was this word invented? When do we use this term? We are yet to make a distinction between a rhino moving at 5m/s and a hummingbird moving at 5m/s. Thus far, how have we handled forces that are only briefly applied such as collisions? (we pretended that doesnt happen) Some believed that this quantity is conserved in our universe.

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H OW IS MOMENTUM RELATED TO OTHER PHYSICS CONCEPTS THAT WE HAVE ALREADY STUDIED ? We will soon see that it has many things in common with Energy, Newtons 3 rd law, and The Calculus. The time rate of change of linear momentum of a particle is equal to the net force acting on the particle.

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P AUSE TO THINK ABOUT CALCULUS CONCEPTS : Why is a derivative involved? What does this say about the slope of a momentum-time graph? The area under which graph might be meaningful? So, how might an integral be involved? Momentum may be changing non-uniformly with time The slope of a momentum-time graph is net force! The area under a force-time graph is a change in momentum! The integral of force with respect to time is a change in momentum!

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P AUSE TO THINK ABOUT CALCULUS CONCEPTS : The integral of force with respect to time is a change in momentum! We call the left- hand side of this equation the IMPULSE of the force

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P AUSE TO THINK ABOUT CALCULUS CONCEPTS : The slope of a momentum-time graph is net force! The area under a force-time graph is a change in momentum or an impulse

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I MPULSE -M OMENTUM T HEOREM : The impulse of a force F equals the change in momentum of the particle. This is another way of saying that a net force must be applied to change an objects state of motion. Why does this look different from the last equation? Because the force might be constant!

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A FEW THINGS ABOUT IMPULSE: It is a vector in the same direction as the change in momentum. It is not a property of an object! It is a measure of the degree to which a force changes a particles momentum. We say an impulse is given to a particle. What are its units? From the equation we see that they must be the same as momentums units (kgm/s). Impulse approximation: assume the force is applied only for an instant and that it is much greater than other forces present.

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A NOTHER QUESTION PLEASE …

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T O STOP A SPEEDING TRAIN : E XPLAIN THESE VIDEOS IN PHYSICS TERMS.

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Q UICK C ONCEPTUAL Q UIZ Can a hummingbird have more momentum than a rhino? Why might an out of control truck hit a haystack or barrels and pile of sand as opposed to a wall as an emergency stop? How is a ninjas ability to break stacks of wood related to impulse and momentum? What good is it to know an objects momentum?

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Question 2: If a boxer is able to make his impact time 5x longer by riding with the punch, how much will the impact force be reduced? By 5x

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When a dish falls, will the impulse be less if it lands on a carpet than if it lands on a hard floor? No – the same impulse – the force exerted on the dish is less because the time of momentum change increases.

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E XAMPLES Examples of Increasing Impact Time to decrease Impact Force: Bend knees when jumping Gymnasts and wrestlers use mats Glass dish falling on carpet rather than concrete Acrobat safety net Other examples???

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O BSERVING CHANGES IN MOMENTUM :

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C ONSIDER TWO PARTICLES THAT CAN INTERACT, BUT ARE OTHERWISE ISOLATED FORM THEIR SURROUNDINGS. What do we know about a collision between these two particles? Newtons law says that they exert equal and opposite forces on each other regardless of comparative size (mass). Is it possible for one particle to be in contact with the second particle for a longer period of time than the second on the first? No, so the impulse imparted on each must be the same. THEREFORE… No, so the impulse imparted on each must be the same. THEREFORE…

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T HE PARTICLES MUST UNDERGO THE SAME CHANGES IN MOMENTUM ! Lets look at this mathematically.

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What does it mean, conceptually, for a time derivative of momentum to be zero It means that the total momentum of the system is constant over time. aka Momentum is Conserved!

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T HE L AW OF C ONSERVATION OF M OMENTUM When two isolated, uncharged particles interact with each other, their total momentum remains constant. OR The total momentum of an isolated system at all times equals its initial momentum (before and after collisions).

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F IND THE REBOUND SPEED OF A 0.5 KG BALL FALLING STRAIGHT DOWN THAT HITS THE FLOOR MOVING AT 5 M / S, IF THE AVERAGE NORMAL FORCE EXERTED BY THE FLOOR ON THE BALL WAS 205N FOR 0.02 S.

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A) v/3 to the left B) The piece is at rest. C) v/4 to the left D) 3v/4 to the left E) v/4 to the right A mass m is moving east with speed v on a smooth horizontal surface explodes into two pieces. After the explosion, one piece of mass 3m/4 continues in the same direction with speed 4v/3. Find the magnitude and direction for the velocity of the other piece.

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H OW GOOD ARE BUMPERS ? A car of mass 1500kg is crash-tested into a wall. It hits the wall with a velocity of -15m/s and bounces off with a velocity of 2.6m/s. If the collision lasts for 0.15s, what is the average force exerted on the car?

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T YPES OF C OLLISIONS Energy is always conserved but may change types (mv 2 /2, mgh, kx 2 /2 etc). There is only one type of momentum (mv). We identify collisions based upon their conservation of kinetic energy. Inelastic kinetic energy is NOT constant Elastic kinetic energy IS constant

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I NELASTIC C OLLISIONS These collisions are considered PERFECT when the objects collide and combine to move as one object. Inelastic Objects bounce but may be deformed so kinetic energy is transformed. Perfectly Inelastic Objects stick together

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P ERFECTLY I NELASTIC C OLLISIONS :

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E LASTIC C OLLISIONS ( IDEALLY )

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F OR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR RELATIVE SPEED OF THE OBJECTS BEFORE AND AFTER COLLISION. From momentum conservation…

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F OR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR FINAL SPEED IN TERMS OF INITIAL SPEEDS AND MASS. From kinetic energy conservation… Divide out ½ and move like mass terms to the same side so mass can be factored out… Factor difference of squares…

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Combine our two results… The relative speed of the two objects before an elastic collision equals the negative of their relative speed after.

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S OLVE FOR FINAL SPEEDS IN TERMS OF INITIAL SPEEDS AND MASS.

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T WO - DIMENSIONAL C OLLISIONS Set coordinate system up with x-direction the same as one of the initial velocities Label vectors in a sketch Write expressions for components of momentum before and after collision for each object v 1i v 1f v 2f v 2f cos φ v 1f sin θ -v 2f sin φ φ v 1f cos θ θ

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T HE TYPES OF COLLISIONS ARE TREATED THE SAME MATHEMATICALLY.

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