This preview shows page 1. Sign up to view the full content.
Unformatted text preview: MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Fall Term 2010 Problem Set 8: Continuous Mass Transfer
Due: Tuesday Nov 2 at 9 pm Place your solutions in the appropriate section slot in the box outside 26-152. Write your name, section, table and group number on the upper right side of your first page. Website: http://web.mit.edu/8.01t/www/ Textbook: Young and Friedman, University Physics Twelfth Edition Oct 25/26 W08D1 Continuous Mass Flow Reading Assignment: Young and Freedman: 8.6, Class Notes: Continuous Mass Flow Oct 27/28 W08D2 Rockets Reading Assignment: Young and Freedman: 8.6; Experiment 3: Impulse, Class Notes: Continuous Mass Flow Oct 29 W08D3 Quiz 6 Integrals, Dot and Cross Products; Group Problem Solving: Falling Chain Nov 1/2 W09D1 Two-Dimensional Rotational Kinematics Reading Assignment: Young and Freedman: 9.1-9.6, 10.5 Nov 3/4 W09D2 Angular Momentum and Two-Dimensional Rotational Dynamics Reading Assignment: Young and Freedman: 10.1-10.2, 10.5-10.6; 11.1-11.3 Nov 5 W09D3 Quiz 7 Momentum; Group Problem Solving: Two Dimensional Rotational Dynamics Problem 1 A rocket has a dry mass (empty of fuel) mr ,0 = 2 ! 107 kg , and initially carries fuel with mass m f ,0 = 5 ! 107 kg . The fuel is ejected at a speed u = 2.0 ! 103 m " s-1 relative to the rocket. What is the final speed of the rocket after all the fuel has burned? Problem 2: An ice skater of mass m is holding a bag of sand of mass ms that is leaking sand at a constant rate b . The ice skater is pushed with a constant force of magnitude F on a frictionless ice surface. At the instant that all the sand has leaked out, what is the speed of the skater? Problem 3: Falling Raindrop A raindrop of initial mass m0 starts falling from rest under the influence of gravity. Assume that the raindrop gains mass from the cloud at a rate proportional to the momentum of the raindrop, dmr / dt = kmr vr , where mr is the instantaneous mass of the raindrop, vr is the instantaneous speed of the raindrop, and k is a constant with units [m -1 ] . You may neglect air resistance. a) Derive a differential equation for the speed of the raindrop. b) Based on your result from part a) (you do not need to integrate your differential equation), show that the speed of the drop eventually becomes effectively constant and give an expression for the terminal speed. Problem 4: Fire Hydrant Water shoots out of a fire hydrant having nozzle diameter D with nozzle speed V0 . What is the reaction force on the hydrant? Problem 5: Rocket in a Constant Gravitational Field A rocket ascends from rest in a uniform gravitational field by ejecting exhaust with constant speed u relative to the rocket. Assume that the rate at which mass is expelled is given by dm f / dt = ! mr , where
mr is the instantaneous mass of the rocket and ! is a constant. The rocket is retarded by air resistance with a force F = bmr vr proportional to the instantaneous momentum of the rocket where b is a constant and vr is the speed of the rocket. a) Derive a differential equation for the speed of the rocket. b) Using the fact that dm f / dt = ! dmr / dt , determine an expression for the mass of the rocket as a function of time. c) Determine u( dm f / dt ) , called the thrust. d) Challenge: determine the speed of the rocket as a function of time. Problem 6 A spacecraft is launched from an asteroid (which we take to be stationary) by bombarding the spacecraft by a beam of rock dust. The beam of dust is ejected from the dust gun at a speed u with respect to the asteroid. The dust is ejected at a constant rate dme b= = !u where ! " b / u is the mass per unit length in the dust beam. Assume that dt the dust comes momentarily to rest at the spacecraft and then slips away sideways; the effect is to keep the spacecraft’s mass ms constant. a) Derive an equation for the acceleration dvs / dt of the spacecraft at time t, in terms of the rate that the dust mass hits the surface of the spacecraft dmd / dt , the speed of the dust relative to the asteroid u , the mass of the spacecraft ms , and the velocity of the spacecraft vs . Show your momentum flow diagrams at time t and time t + !t . Clearly identify your system and label all the objects in your system. What is the terminal velocity of the spacecraft? Hint: dmd / dt ! b . b) Using conservation of mass, show that the rate that the dust mass hits the spacecraft, dmd b = ( u ! vs ) dt u
where vs is the speed of the spacecraft, the rate that the dust mass is shot from the asteroid dme / dt = b , and the speed u of the dust relative to the asteroid. c) Use your results from part b) in part a) to find a differential equation for the speed vs of the spacecraft. d) What do you expect is the terminal speed of the spacecraft? e) Challenge problem: Find the speed vs of the spacecraft as a function of time, assuming vs (t = 0) = 0 by integrating your differential equation in part c). (If you get an integral that you are not sure how to integrate, you can leave your answer in integral form.) ...
View Full Document