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Lecture #2 Questions on Assignment #1? Homework Expectations /Office hours Impulse and momentum conservation Rocket propulsion Worked problems Cross-product problem Impulse and Momentum V-5-7 1 :10 Homework Bliss GOOD NOT GOOD mr mass of cannon 2.1103 kg m1 2.1103 mc mass of chicken 2.4 kg m2 2.4 J cannon impulse (1000 Nt s ) xˆ impulse 1000 vic , v fc initial , final velocity of chicken vi , v f mcannon 2.110 kg ; mchicken 2.4kg 3 vi chicken , v f chicken Brief description # of points No attempt Just started / confused Well along Nearly perfect Physics Bliss 0 4 7 9 10 2 :17 Definition of Impulse J F (t ) t J t2 F (t ) dt Impulse J is a useful concept in the study of collisions. (e.g. Balls and bats, automobiles, comets and planets) Impulse is the average force acting W F ( r ) r over a time period multiplied by r2 the time period. W F ( r ) dr r1 It may also be written as an integral Note the difference between impulse J and work W. t1 3 :22 Definition of Impulse J F (t ) AV t t2 J F (t )dt t1 W FAV s r2 W F d r1 Impulse J is a useful concept in the study of collisions. (e.g. Balls and bats, automobiles, comets and planets) Impulse is the average force acting over a time period multiplied by the time period. It may also be written as an integral Note the difference between impulse J and work W. 4 :22 Impulse and momentum change t2 J F (t )dt t1 J t2 t1 dp(t ) dt p (t2 ) p (t1 ) p dt Two equivalent impulses with different Fmax 1 Force (N) 0.8 0.6 0.4 0.2 0 0 2 4 6 Time (s) 8 10 Impulse is useful because it directly allows expression of momentum change. 5 :27 Impulse I -- Problem #L2-2 “A car crash” Jack and Jill were ingesting things they shouldn’t while driving two cars of mass 1000 kg and 2000 kg with velocity vectors 30 x m/s and 10 x 60 y m/s We may look at their collision in terms of impulse. Jill’s car applied an impulse to Jack’s car as follows: J (13.33x ˆ 40 yˆ ) 103 Nt s What is the final momentum of Jack’s car only? What is his final velocity? Calculate pJack final , vJack final and vJack final 6 :32 Rocket Science I F ma Not the whole story dP More general – get used to it F dt Case where m not constant d F mv dt v) (mv m vexhaust defined as " thrust" m marocket thrust Instantaneous acceleration, but rocket keeps getting lighter!! 7 :40 Rocket Science II mr v ve m In absence of other forces v m f dm dv ve Separable ODE. 0 mi m Integrate both sides mf v f ve ln Rocket velocity in mi terms of lost mass m f and exhaust velocity v f 2.3ve log 10 mi 8 :45 Rocket Science III mf v f 2.3ve log 10 mi If a rocket is 60% fuel, then it limits at ve If a rocket is 90% fuel, then limit at 2.3 ve At 99% fuel, limit is 4.6 ve Achieving escape velocity with Single-Stage to Orbit rocket 70000 Exhaust velocity 3350 mph Exhaust velocity 6700 mph Exhaust velocity 13400 Escape velocity 65000 60000 55000 50000 Velocity (mph) 45000 40000 35000 30000 25000 20000 15000 10000 5000 0 90 80 70 60 50 40 30 Percentage of rocket mass remaining 20 10 1 9 :50 Specific Impulse I sp Engine Type Best Chemical Best Nuclear Xenon Ion Linear Accel. EM Catapult ve g Specific Impulse 500 2,000 3,800 1,000,000 N/A In practice of rocketry, the “specific impulse” is often quoted Exhaust Velocity m/s mph 4,900 10,363 19,600 41,452 37,240 78,759 9,800,000 20,726,119 45,000,000 95,170,955 Rocket Vfinal mph 23,835 95,340 181,146 47,670,074 95,170,955 Travel to Alpha-C (yrs) 106,477 26,619 14,010 53 27 10 :45 :55 XIPS Xenon Ion Propulsion System Thrusts are 60-200 milliNewtons Used for “station-keeping” and deepspace missions 11 :45 :55 Problem T3.7 The first couple of minutes after a space shuttle launch can be described as follows: The initial mass is 2x10E6 kg, the final mass (after 2 minutes) is about 1x10E6 kg, the average exhaust speed is about 3000 m/s. If all this were taking place in outer space, with negligible gravity, what would be the shuttle’s speed at the end of this stage? What is the thrust during this same period and how does it compare with the total initial weight of the shuttle (on earth)? 12 :65 Impulse II -- Problem #L2-3 “Another car crash” Jack and Jill are partly recovered from their previous injuries, and haven’t learned from their experience. They are drinking cokes with Bacardi 151 and not wearing seat-belts. Jack’s vehicle has velocity vector 30 xˆ m / s 30 xˆ m / s Jill’s vehicle has Both vehicles’ mass=M. Both people’s mass=70 kg. Solve for case of inelastic and elastic collisions of vehicles. Jill has an airbag in her vehicle. It takes her 100 millisec to reduce her velocity relative to her vehicle to zero. Jack stops 5 millisec after impacting the steering wheel. What impulse does each person experience? What is 13 average force for each? How many “g’s” do they feel.:75 Lecture #2 Wind-up . Impulse J t F (t )dt p t . mv exhaust defined as " thrust " . mf v f ve ln mi Got on WebCT? / Got Books? Office hours Wednesday 4-5:30 Homework problems in Taylor, Handout, Galileo problem – check the web. Second homework due in class Thursday 9/4 2 1 (Includes introducing gravity into rocket equation) 14 :72