A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (Including its fuel) is m, the fuel is consumed at rate r,and the exhaust gases are ejected with constant velocity v. (relative to the rocket). A model for the velocity of the rocket at time t is given by the equationmv(t) = -gt - v, Inmwhere g is the acceleration due to gravity and t is not too large. If g = 9.8 m/s2, m = 27,000 kg, r = 160 kg/s, and v = 3,100 m/s, find the following.(a) the height of the rocket (in meters) one minute after liftoff (Round your answer to the nearest meter.)(b) the height of the rocket (in meters) after it has consumed 5,000 kg of fuel (Round your answer to the nearest meter.)m

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Answered: A rocket accelerates by burning its…

University Physics Volume 3

17th Edition

ISBN:9781938168185

Author:William Moebs, Jeff Sanny

Publisher:William Moebs, Jeff Sanny

Chapter5: Relativity

Section: Chapter Questions

Problem 55P: (a) Find the momentum of a asteroid heading towards Earth at 30.0 km/s. (b) Find the ratio of this...

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4. A rocket accelerates by burning its onboard fuel, so that its mass decreases with time.Suppose that the initial mass of the rocket at liftoff (including its fuel) is m, the fuel isconsumed at rate r, and the exhaust gases are ejected with constant velocity ν (relativeto the rocket). A model for the velocity of the rocket at time t is given by the equationv(t) = −ν ln(1 −Rt) −gt,where g is the acceleration due to gravity, R = r/m, and t is not too large.(a) Find an expression for the acceleration, a(t) = v′(t), of the rocket at any time t.(b) Find an expression for the position of the height, h(t), of the rocket at any timet. You can assume that at t = 0, the rocket hasn’t left the ground. Hint: re-member that ν,R,g are all constants and t is the independent variable.Write out your steps carefully!(c) Let g = 9.8 m/s2, R = 0.005 s−1. What does the velocity of the exhaust gases,ν, need to be for the rocket to reach a height of 6000 m one minute after liftoff?You can round to the…
The following equation describes the motion of a rocket that is expelling a total mass M of fuel at a rateα = -dm/dt:dv/dt = uα/(M-αt),where u is the speed the fuel is ejected, v is the velocity of the rocket in the x-direction, and t is time. Assume the velocity at t = 0 is v(0) = v0, and that M, and u are constants. The burn rate is proportional to the mass of the rocket, as follows:α = -βm,where m is the current mass of the rocket (which is therefore time-dependent). This means that as less fuel is available to burn, then the fuel is burned at a lower rate. Write an expression for the mass of the fuel as a function of time. Now, write an expression for the velocity as a function of time for the rocket. A particular rocket burns fuel at a rate proportional to the mass of the fuel with the constant of proportionality being 6.4 1/s. The propellant speed is 5.1 m/s. The total mass of fuel is 136.4 kg. What is the speed, in meters per second, of the rocket at t = 2.7 s?
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