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Announcements 7 Oct Prayer 2. Energy review E before + W = E after Conservation of energy W = F // d KE = ½ mv 2 PE gravity (near surface of Earth) = mgy y is measured relative to where you define
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Announcements 7 Oct Prayer 2. Energy review E before + W = E after Conservation of energy W = F // d KE = ½ mv 2 PE gravity (near surface of Earth) = mgy y is measured relative to where you define y=0 PE gravity (otherwise) = -GmM/r r is measured from center to center PE spring = ½ kx 2 x is measured from equilibrium (relaxed) position of spring Also remember F spring = kx, if needed in a N2 problem Colton - Lecture 11 pg 1 W includes work by all nonconservative forces E includes both kinetic and potential energies of all objects Which of the problems from last night's HW assignment would you most like me to discuss in class today? Colton - Lecture 11 pg 2 Worked problem How much energy would you have to provide in order to shoot a 100 kg satellite into a near orbit like the ISS, 6712 km from center of earth? (E.g. via initial KE) Answer: 3.29E9 J Colton - Lecture 11 pg 3 Colton - Lecture 11 pg 4 Escape velocity How fast do you have to shoot an object to get it infinitely far away form the Earth? R earth = 6371 km; M earth = kg (ignore the sun s gravitational pull) Compare to orbital velocity Robert Heinlein: If you can get into orbit, then you're halfway to anywhere Answers: 11.2 km/s; 7.9 km/s Colton - Lecture 11 pg 5 Colton - Lecture 11 pg 6 Power! The rate at which energy is produced or consumed Or (equivalently) Power is the rate at which work is being done SI units: 1 Watt = 1 J/s = 1 N m/s = 1 kg m 2 /s 3 (yuck!) For reference: 1 horsepower (hp) = W Colton - Lecture 11 pg 7 Another useful formula, if constant velocity: P = F // v Colton - Lecture 11 pg 8 Image credit: Wikipedia. (Northern end of Stelvio Pass, Italy) From warmup: Switchbacks on mountain roads (consider only work done against gravity): a. increase the work needed to go up a mountain b. decrease the work needed to go up a mountain c. keep the work needed the same Colton - Lecture 11 pg 9 Experimental Problem How much horsepower can a person generate? Experiment: jumping from a stand-still Volunteer needed! Parameters: mass (kg) = measured height jumped (m) = measured impulse time (s) (time while legs are exerting force on ground) = What was the work done by his/her body during the impulse time? How much horsepower? Colton - Lecture 11 pg 10 Clicker quiz A car weighing 3000 N moves at a speed of 30 m/s on level ground. To do this, it pushes backwards on the road with a 5000 N force. What is the power output of the car engine? a. 0 kw b. 60 kw c. 90 kw d. 150 kw e. 240 kw Colton - Lecture 11 pg 11 Where are we now? Topics Kinematics (velocity, acceleration) Vectors & 2D Motion Forces & Newton s Laws Work & Energy Momentum Rotations, Torque, and Angular Momentum Pressure Fluids & Solids Temperature, Heat, and Heat Flow Laws of Thermodynamics Vibrations & Waves Mechanics Thermodynamics Part Mechanics, Part Sound Part Optics Colton - Lecture 11 - pg 12 Conserved quantities Energy When no non-conservative work done, E bef = E aft Mass If not converted to/from energy (E=mc 2 ), (total mass) bef = (total mass) aft Charge (total charge) bef = (total charge) aft I.e., if some positive charge flows out of a neutral object, it will leave the object with negatively charged Often conserved (used to e.g. balance chemical reactions) Number of each type of atom Number of electrons Etc. Colton - Lecture 11 - pg 13 A new conserved quantity momentum Define p mv for each object, then p before p after Another blueprint equation! (if no external forces) Careful: Momentum is a vector! Colton - Lecture 11 - pg 14 Momentum: used for Collision Problems v 1 initial v 2 initial m 1 m 2 Colton - Lecture 11 - pg 15 Derivation of conservation law: F 2-1 F 1-2 F ma F m a Newton s 3 rd Law: the forces involved in the collision are and If no other forces, then F F 1-2 = m 1 a 1 + m 2 a 2 0 = m 1 v 1 /t + m 2 v 2 /t Multiply by t (which is the same for both) m 1 v 1 + m 2 v 2 = 0 m 1 (v 1 final - v 1 initial ) + m 2 (v 2 final - v 2 final ) = 0 m 1 v 1 initial + m 2 v 2 initial = m 1 v 1 final + m 2 v 2 final and there you have it! Colton - Lecture 11 - pg 16 From warmup The total momentum of an isolated system of objects is conserved a. only if conservative forces act between the objects b. regardless of the nature of the forces between the objects. Colton - Lecture 11 - pg 17 Why use conservation of momentum? Colton - Lecture 11 - pg 18 Demo Problem: A cart moving at 1 m/s runs into a second cart (stationary) with the same mass and sticks to it. What velocity do the two stuck together carts now have? Demo Problem: A cart moving at 1 m/s runs into a second cart (stationary) with twice the mass and sticks to it. What velocity do the two stuck together carts now have? Colton - Lecture 11 - pg 19 Demo Problem: A cart moving at 1 m/s runs into a second cart with twice the mass and sticks to it. The second cart is moving at 0.5 m/s towards the first one. What velocity do the two (stuck together) carts now have? Colton - Lecture 11 - pg 20 Dr Colton s Guide: How to solve Conservation of Momentum problems 1. Draw initial and final pictures 2. Draw momentum or velocity vectors (arrows) in each picture 3. Use p before pafter as blueprint equation 4. Divide into separate x- and y- equations if needed 5. Fill in both sides of blueprint equation(s) using initial and final pictures: one term in equation for each arrow in picture. 6. Reminder: be careful with signs! (Momentum is a vector) Colton - Lecture 11 - pg 21 The new blueprint p p before after (if no external forces) Compare to previous two blueprint equations: F E before ma E after (if no non-conservative forces) Similarities? Differences? Colton - Lecture 11 - pg 22 From warmup Suppose Ralph is floating in outer space with no forces acting on him. He is at rest, so his momentum is zero. Now, he throws a ball. The ball goes one way, and he goes the other way. Before the collision, there was no momentum, and after the collision, there is plenty of momentum! Was momentum conserved? Think-pair-share Think about it for a bit Talk to your neighbor, find out if he/she thinks the same as you Be prepared to share your answer with the class if called on Clicker: I am now ready to share my answer if randomly selected. a. Yes Note: you are allowed to pass if you would really not answer. Colton - Lecture 11 - pg 23 Worked Problem In the new sport of ice football, a 100 kg defensive end running north at 4 m/s tackles a 75 kg quarterback running east at 7 m/s. There s no friction. What is their combined velocity right after the tackle? Answers: v x = 3 m/s; v y = 2.28 m/s; v = 3.77 at 37.3 north of east Colton - Lecture 11 - pg 24 Worked Problem An artillery shell of mass 20 kg is moving east at 100 m/s. It explodes into two pieces. One piece (mass 12 kg) is seen moving north at 50 m/s. What is the velocity (magnitude and direction) of the other piece? Answers: v x = 250 m/s; v y = -75 m/s; v = 261 m/s at 16.7 south of east Colton - Lecture 11 - pg 25 Limitation Like conservation of energy, conservation of momentum is a before and after law which doesn t tell you about: If you want to know about, you have to know Colton - Lecture 11 - pg 26 Impulse Concept: A force exerted for a brief time causes a change in momentum Ft p Impulse equation (focusing on one object) Derivation: F ma v F m t Ft mv Ft p F When to use? Colton - Lecture 11 - pg 27 Worked Problem Two steel balls (same mass, m = kg) bounce off of each other. High speed photography reveals that the two balls are in contact for s. Before the collision, the left ball is traveling to the right at 2 m/s and the right ball is traveling to the left at 2 m/s. After the collision, the left ball is traveling to the left, still at 2 m/s, and the right ball is traveling to the right at 2 m/s. What was the contact force during the collision? Answer: N Colton - Lecture 11 - pg 28 From warmup, do as clicker quiz A ping-pong ball moving forward with a momentum p strikes and bounces off backwards from a heavier tennis ball that is initially at rest and free to move. The tennis ball is set in motion with a momentum: a. greater than p b. less than p c. equal to p What about if ping-pong ball thuds and falls flat? Demo: Elastic and Inelastic Pendulum which will cause the wood to be knocked over? Colton - Lecture 11 - pg 29
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