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Momentum & Collisions
1.
AP Physics Rapid
Learning Series - 08 Rapid Learning Center Chemistry :: Biology :: Physics :: Math Rapid Learning Center Presents …… Teach Yourself AP Physics in 24 Hours 1/49 *AP is a registered trademark of the College Board, which does not endorse, nor is affiliated in any way with the Rapid Learning courses. Momentum and Collisions Physics Rapid Learning Series 2/49 Rapid Learning Center www.RapidLearningCenter.com/ © Rapid Learning Inc. All rights reserved. Wayne Huang, Ph.D. Keith Duda, M.Ed. Peddi Prasad, Ph.D. Gary Zhou, Ph.D. Michelle Wedemeyer, Ph.D. Sarah Hedges, Ph.D. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 1
2.
AP Physics Rapid
Learning Series - 08 Learning Objectives By completing this tutorial, you will: „ Understand the concepts of momentum and impulse. „ Mathematically describe various types of collisions. „ Apply these ideas to 2- 3/49 2 dimensional collisions. Concept Map Physics Studies Previous content New content Motion Caused by collisions Forces F Elastic lli i creates Momentum is a Conserved vector Applied over time gives Impulse And quantity Vectors Inelastic collisions 4/49 08 : 3/32 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 2
3.
AP Physics Rapid
Learning Series - 08 Momentum Although only moving objects possess momentum its concept is very similar to 5/49 momentum, inertia. Momentum Definition In a way, momentum is the motion equivalent of inertia. A large moving object is more difficult to stop than a small moving object. 6/49 Momentum describes or quantifies this tendency to keep moving. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 3
4.
AP Physics Rapid
Learning Series - 08 Momentum Formula Momentum is the product of mass and velocity. P = mv momentum, t kg•m/s mass, kg velocity, m/s 7/49 Substantial Momentum A bullet may have a large momentum although its mass is small. P= mv A tanker ship has a large momentum although its velocity is small. P 8/49 P= mv © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 4
5.
AP Physics Rapid
Learning Series - 08 Changing Momentum Obviously, all moving objects can be stopped eventually . Thus, their momenta are changed. This is usually accomplished by changing the velocity of an object. 9/49 Changing Momentum with Force To change the momentum of any object, you must apply a force to it. 10/49 This force must be applied over a period of time. This could be a very short period of time, or a very long period of time. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 5
6.
AP Physics Rapid
Learning Series - 08 Impulse Formula Impulse is equal to the product of force and time. J = Ft I l k / F N ti 11/49 Impulse, kg•m/s Force, time, s Impulse Observations „ Impulse can also be defined as a change in momentum, ΔP. „ Notice that the units of momentum and impulse are the same (kg•m/s). „ This makes sense since impulse is just a change in the amount of momentum. 12/49 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 6
7.
AP Physics Rapid
Learning Series - 08 Impulse and Momentum Formula Since impulse is defined as a change in momentum, it can also be written as: FΔt = Δ(mv) Delta, change When an object’s 13/49 in, time momentum changes, it is usually the velocity that changes, not the mass. No Bounce Interaction A ball is falling towards the ground. It would take some change in momentum, or impulse, to stop a moving object. P = mv moving Hits ground Since v =0 P = 0 ΔP = mv 14/49 g ball not moving © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 7
8.
AP Physics Rapid
Learning Series - 08 Bouncing Interaction However, it would take even more impulse to stop the object, and then make it ricochet in the opposite direction. P = -mv ball moving Ball bouncing up Hits ground P = +mv ΔP = 2mv Twice as much 15/49 down as before Impulse Example During a rainstorm, drops come straight down with a velocity of -15 m/s and hit a car roof. The mass of rain hitting the car is .060 kg/s. Assuming the rain comes to rest when it hits the car, how much force is exerted on the car roof by the rain? 16/49 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 8
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AP Physics Rapid
Learning Series - 08 Solution Begin from our impulse formula. FΔt = Δ(mv) F = mΔ v Δt Change in 17/49 F m(vf vi ) − Δt = velocity, Δv Calculation F .060 kg ( 0 m/s 15m/s) 1sec − − = F = .060 kg ( 15m/s) 1sec two – signs will F = .9 kgm F = .9N s2 2 yield a + 18/49 This is the force on the RAIN. The force on the car must be in the opposite direction ( Newton’s 3rd law). F= -0.9 N © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 9
10.
AP Physics Rapid
Learning Series - 08 Additional Question If the raindrops were frozen with the same mass and velocity, if they hit the car and ricocheted upwards, how would that affect the force on the car? Because they would be changing their momentum more, due to the larger change in their velocity, a greater t i impulse l would ldb be required. Thus, more force would be applied to the car. 19/49 Conservation of Momentum In any type of collision or interaction, the conservation of momentum will be a key 20/49 idea to understanding the situation. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 10
11.
AP Physics Rapid
Learning Series - 08 Conservation Laws In all collisions or interactions, the momentum of a system is always conserved. You may have previously learned about conservation of mass or energy from chemistry class. 21/49 Vector Quantity Momentum is a vector quantity, direction must be taken into account to see that momentum truly is conserved. 22/49 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 11
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AP Physics Rapid
Learning Series - 08 Rifle and Bullet Example The rifle and bullet can be considered a system. Before firing, they are both motionless and have a total momentum of 0. After firing, the total momentum still equals 0. The rifle has momentum to the left, the bullet to the right. The rifle has a much larger mass so its velocity is less, but their momentum is still conserved. 23/49 mv mv Additional Observations You may recall that Newton’s 3rd Law fits this example too. The force on the rifle is equal and opposite to the force on the bullet. Reaction Force Action Force However, due to the difference in mass, the 24/49 acceleration of the bullet is much greater than the acceleration of the rifle. Acceleration is not a conserved quantity. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 12
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AP Physics Rapid
Learning Series - 08 Conservation Problems When solving problems involving the conservation of momentum, the most important thing to consider is: Total momentum b f lli i Total momentum ft lli i = 25/49 before collision after collision Explosion Sample Problem A 300 kg cannon fires a 10 kg projectile at 200 m/s. How fast does the cannon recoil backwards? BOOM 26/49 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 13
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AP Physics Rapid
Learning Series - 08 Solution Set up The momentum of the projectile must be equal in size to the momentum of the cannon. They must be equal since they must cancel each other out. Pbefore = Pafter 27/49 BOOM Calculation Pafter = Pbefore mcannonvcannon + mprojvproj = 0 Before firing, velocity = 0m/s. cannon cannon proj proj (300 kg) (vcannon) + (10kg) (200m/s) = 0 v 2000kgm/s cannon 300k − = 28/49 300kg vcannon = -6.67 m/s Negative sign indicates the cannon moves in the opposite direction to the projectile © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 14
15.
AP Physics Rapid
Learning Series - 08 Hit and Stick Sample Problem Joe has a mass of 70 kg and is running at 7 m/s with a football. He slams into 110 kg Biff who was initially motionless. During this collision, Biff holds onto and tackles Joe. This type of event may be called a “hit and stick” collision. What is their resulting velocity after the collision? 29/49 Hit and Stick Solution Pbefore = Pafter PJoe PBiff PJoe+Biff + = Biff’s initial velocity is zero, so this term drops out. m1v1 +m2v2 = (m1 +m2 )v3 m1v1 = (m1 +m2 )v3 70kg(7m/s) = (70kg+110kg)v3 Since they stick together, add their masses. Do math 30/49 v3 70kg(7m/s) = (70kg + 110kg) v3 2.7m/s = carefully Since all velocities were in the same direction, no – signs are needed here. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 15
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AP Physics Rapid
Learning Series - 08 Hit and Rebound Sample Problem A 1 kg basketball rolls at +5 m/s and collides with a stationary 4 kg bowling ball. The bowling ball is given a velocity of +2 m/s. What is the velocity of the basketball after the collision? Find Vbasketball after collision? 31/49 Hit and Rebound Solution Pbefore = Pafter v=0m/s here Pbasket +Pbowling = Pbasket +Pbowling D Do math th carefully m1v1 +m2v2 =m1v3 +m2v4 m1v1 =m1v3 +m2v4 (1kg)5m/s = (1kg)v + (4kg)2m/s - sign shows the basketball is moving in 32/49 (1kg)5m/s - (4kg)2m/s = (1kg)v the opposite direction (1kg)5m/s - (4kg)2m/s = v = −3m/s (1kg) © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 16
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AP Physics Rapid
Learning Series - 08 Summary of Collisions Explosion: one object breaking into more objects. 0 =mv +mv + ... Hit and stick: one object striking and joining to the other. + = mv1 mv2 (m1 +m2 )v3 33/49 Hit and rebound: one object striking and bouncing off of the other. + = m1v1 m2v2 m1v3 +m2v4 Subscripts In the momentum conservation formulas, various subscripts are often used to keep track of the various objects. Don’t be confused by them, they are used to help keep track of the variables. Some examples: 34/49 m1v1=1st mass and 1st velocity of that mass mAv’=1st mass and the 2nd velocity of that mass m2xv2x=2nd mass and 2nd velocity only in the x direction © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 17
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AP Physics Rapid
Learning Series - 08 Elastic and Inelastic Collisions All collisions or interactions can be described as elastic or inelastic 35/49 collisions. Elastic Collision „ Momentum is conserved. „ The objects colliding aren’t deformed or smashed „ Thus no kinetic energy is lost; kinetic energy is conserved also. „ Ex: billiard ball collisions 36/49 © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 18
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AP Physics Rapid
Learning Series - 08 Inelastic Collision „ Momentum is conserved. „ Kinetic energy is lost. „ The energy may be transformed into sound sound, deformation of materials, flying debris, etc. „ Often objects interlock or stick together. „ Objects are also often deformed or crunched. „ Example: Car crash 37/49 Joe and Biff’s Collision Is our previous football tackle an example of an elastic or inelastic collision? Before Collision After Collision m1 = 70 kg v1 = 7 m/s m2 = 110 kg v2 = 0 m// s m1+2 = 180 kg v3 = 2.7 m/s 38/49 We know momentum is conserved, but if kinetic energy is conserved too, then it would be an elastic collision. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 19
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AP Physics Rapid
Learning Series - 08 Calculation of Kinetic Energy KE mv 2 2 = KE formula from a previous tutorial Before collision: After collision: KE 180kg(2.7m/s) 2 2 KE 70kg(7m/s) 2 2 = = 39/49 KE = 1715J KE = 656J Over 1000 J of energy are lost due to friction, heat, deformation, etc! Definitely an inelastic collision. Additional Directions Many collisions involve motion in more than one direction. The same concepts 40/49 can be applied to describe these new situations. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 20
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AP Physics Rapid
Learning Series - 08 2-Dimensional Collisions Collisions do not always take place in a nice neat line. Often, collisions take place in 2 or 3 dimensions: 41/49 Although the mathematics needed to show this may be somewhat lengthy, the general idea can easily be conveyed. 2-Dimensional Movement One ball collides into another. By using momentum vector components, you can predict the result: Before impact: After impact: BAM 42/49 Total P before Y components cancel out X components add up to previous P © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 21
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AP Physics Rapid
Learning Series - 08 Consider the Components It’s easiest to break the momentum into X and y components. Since momentum is always conserved: PX before =PX after P P 43/49 PY before = PY after Sample 2-Dimensional Problem Two pool balls, each 0.50 kg collide. Initially, the first moves at 7 m/s, and the second is motionless. After the collision, the first moves 40o to the left of its original direction, the second moves 50o to the right of its original direction. Find the velocitiy of B after the collision. A B A After Collision 40o 44/49 Before Collision B 50o © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 22
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AP Physics Rapid
Learning Series - 08 Consider the Components The X and Y components of momentum are both conserved. You can visualize this several ways: A A B B 45/49 After the collision, the sum of the X components equals the original momentum. The y components cancel out since there was no momentum in that direction originally. Add the Vectors Without using components, it can also be noticed that both momentum vectors after the collision add up to the original momentum vector: A B A Resultant equal to Pbefore 46/49 B Remember that vectors can be moved anywhere as long as their magnitude and relative direction are unchanged. They are added tip to tail. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 23
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AP Physics Rapid
Learning Series - 08 Problem Solution sinθ = opp hyp sin 40o = PB P Use trig to find the momentum of ball B. Then find its velocity. A B 40o 3.5 kgm/s PB 2.25 kgm/s = 47/49 Pbefore = mv = = .5kg(7m/s) 3.5kgm/s P = m v B B B 2.25 kgm/s = .5 kg v v 4.5 m/s B B = Only the Vectors are Conserved When observing the conservation of momentum, be sure to remember that momentum is a vector quantity. Adding the values as scalars won’t work. The direction must be accounted for by vector 48/49 diagrams, trigonometry, etc. © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 24
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AP Physics Rapid
Learning Series - 08 Learning Summary Collisions b l ti Momentum = mass x All momentum concepts can also be may be elastic or inelastic velocity applied to more than 1 dimension 49/49 In any collision, momentum is conserved Impulse = Force x time Congratulations You have successfully completed the core tutorial Momentum and Collisions Rapid Learning Center © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 25
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AP Physics Rapid
Learning Series - 08 Rapid Learning Center Chemistry :: Biology :: Physics :: Math Wh What’s t’ Next N t … Step 1: Concepts – Core Tutorial (Just Completed) Æ Step 2: Practice – Interactive Problem Drill Step 3: Recap Super Review Cheat Sheet 51/49 – Go for it! http://www.RapidLearningCenter.com © Rapid Learning Inc. All rights reserved. - http://www.RapidLearningCenter.com 26
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