Transcript AP Momentum
THE IMMACULATE RECEPTION MOMENTUM AP Physics C: Mechanics WHAT IS MOMENTUM? 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? WHAT IS MOMENTUM? What is its definition? Momentum: “quantity of motion” -Newton Momentum: “mass in motion” Momentum: the product of an object’s mass and its velocity Momentum: It is a vector! Momentum: is sometimes called linear momentum WHAT IS MOMENTUM? How do we calculate it? If object is moving in arbitrary direction: p mv What are its units? mass length time kgm s p x mv x p y mvy p z mv z WHAT DO WE KNOW ABOUT MOMENTUM? WHAT IS MOMENTUM? 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 doesn’t happen) Some believed that this quantity is conserved in our universe. HOW IS MOMENTUM RELATED TO OTHER PHYSICS CONCEPTS THAT WE HAVE ALREADY STUDIED? dv dp F ma m dt dt The time rate of change of linear momentum of a particle is equal to the net force acting on the particle. We will soon see that it has many things in common with Energy, Newton’s 3rd law, and The Calculus. PAUSE TO THINK ABOUT CALCULUS CONCEPTS: Why is a derivative involved? Momentum may be changing non-uniformly with time What does this say about the slope of a momentum-time graph? The slope of a momentum-time graph is net force! The area under which graph might be meaningful? The area under a force-time graph is a change in momentum! So, how might an integral be involved? The integral of force with respect to time is a change in momentum! PAUSE TO THINK ABOUT CALCULUS CONCEPTS: The integral of force with respect to time is a change in momentum! dp F dt We call the lefthand side of this equation the IMPULSE of the force Fdt dp Fdt dp Fdt p I Fdt p tf ti PAUSE 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 IMPULSE-MOMENTUM THEOREM: The impulse of a force F equals the change in momentum of the particle. I Ft p This is another way of saying that a net force must be applied to change an objects state of motion. Why Because does thisthe force look different might be from theconstant! last equation? 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 momentum’s units (kgm/s). Impulse approximation: assume the force is applied only for an instant and that it is much greater than other forces present. ANOTHER QUESTION PLEASE… TO STOP A SPEEDING TRAIN: EXPLAIN THESE VIDEOS IN PHYSICS TERMS. QUICK CONCEPTUAL QUIZ 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 ninja’s ability to break stacks of wood related to impulse and momentum? What good is it to know an object’s momentum? 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? Ft mv By 5x Ft mv t t mv F t 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. EXAMPLES 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??? OBSERVING CHANGES IN MOMENTUM: CONSIDER TWO PARTICLES THAT CAN INTERACT, BUT ARE OTHERWISE ISOLATED FORM THEIR SURROUNDINGS. What do we know about a collision between these two particles? Newton’s 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… THE PARTICLES MUST UNDERGO THE SAME CHANGES IN MOMENTUM! Let’s look at this mathematically. dp1 F2on1 dt dp 2 F1on 2 dt dp1 dp2 dt dt dp1 dp2 0 dt dt d p1 p2 0 dt d p1 p2 0 dt p tot p1 p 2 dp tot 0 dt 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! THE LAW OF CONSERVATION OF MOMENTUM 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). p1i p 2i p1f p 2f FIND THE REBOUND SPEED OF A 0.5 KG BALL FALLING STRAIGHT DOWN THAT HITS THE FLOOR MOVING AT 5M/S, IF THE AVERAGE NORMAL FORCE EXERTED BY THE FLOOR ON THE BALL WAS 0.02S. I Ft p F N Fg t mv v0 205N 5N0.02s v 0.5kg v 3m/s 5m/s 205N FOR 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. 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 p before p after 3m 4v m mv v 4 3 1 4 2 m mv mv v2 4 HOW 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? I mv v 0 I 1500kg2.6m/s (15m/s) I 2.64 104 kgm/s I Ft 2.64 104 kgm/s F 0.15s F 1.76 105 N TYPES OF COLLISIONS Energy is always conserved but may change types (mv2/2, mgh, kx2/2 etc). There is only one type of momentum (mv). We identify collisions based upon their conservation of kinetic energy. Inelastic Elastic • kinetic energy is NOT constant • kinetic energy IS constant INELASTIC COLLISIONS These collisions are considered PERFECT when the objects collide and combine to move as one object. Inelastic Perfectly Inelastic •Objects bounce but may be deformed so kinetic energy is transformed. •Objects stick together PERFECTLY INELASTIC COLLISIONS: p1i p 2i p12f m1v1 m2v2 m1 m2 vf ELASTIC COLLISIONS (IDEALLY) m1v1i m2 v 2i m1v1f m2 v 2f 1 1 1 1 2 2 2 2 m1v1i m2v2i m1v1f m2v2f 2 2 2 2 FOR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR RELATIVE SPEED OF THE OBJECTS BEFORE AND AFTER COLLISION. From momentum conservation… m1v1i m2 v 2i m1v1f m2 v 2f m1v1i m1v1f m2 v 2f m2 v 2i m1v1i v1f m2 v2f v2i FOR ELASTIC COLLISIONS, FIND AN EXPRESSION FOR FINAL SPEED IN TERMS OF INITIAL SPEEDS AND MASS. From kinetic energy conservation… 1 1 1 1 2 2 2 2 m1v1i m2v2i m1v1f m2v2f 2 2 2 2 Divide out ½ and move like mass terms to the same side so mass can be factored out… 2 m1 v1i v1f 2 m v 2 2 2f v 2i Factor difference of squares… m1v1i v1f v1i v1f m2 v2f v2i v2f v2i 2 m1v1i v1f m2 v2f v2i m1v1i v1f v1i v1f m2 v2f v2i v2f v2i Combine our two results… v1i v1f v2f v2i v1i v2i v1f v2f v1i v2i v1f v2f The relative speed of the two objects before an elastic collision equals the negative of their relative speed after. SOLVE FOR FINAL SPEEDS IN TERMS OF INITIAL SPEEDS AND MASS. m1v1i m2 v 2i m1v1f m2 v 2f 1 1 1 1 2 2 2 2 m1v1i m2v2i m1v1f m2v2f 2 2 2 2 TWO-DIMENSIONAL COLLISIONS 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 v1f before and after collision for each object v1fsinθ v1i v1fcosθ θ φ m1v1i m1v1 f cos m2v2 f cos 0 m1v1 f sin m2v2 f sin -v2fsinφ v2fcosφ v2f THE TYPES OF COLLISIONS ARE TREATED THE SAME MATHEMATICALLY. p p i f