Example 4.12 ROCKET DIRECTED VERTICALLY A small rocket, with an initial mass of 900 kg. is to be launched vertically. Up

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Example 4 12 Rocket Directed Vertically A Small Rocket With An Initial Mass Of 900 Kg Is To Be Launched Vertically Up 1 (46.18 KiB) Viewed 36 times

Example 4 12 Rocket Directed Vertically A Small Rocket With An Initial Mass Of 900 Kg Is To Be Launched Vertically Up 2 (50.25 KiB) Viewed 36 times

Example 4.12 ROCKET DIRECTED VERTICALLY A small rocket, with an initial mass of 900 kg. is to be launched vertically. Upon ignition the rocket consumes fuel at the rate of 5kg/s and ejects gas at atmospheric pressure with a speed of 3500 m/s relative to the rocket. Determine the initial acceleration of the rocket and the rocket speed after 10 s, if air resistance is neglected Given: Small rocket accelerates vertically from rest. Initial mass, M = 400 kg Air resistance may be neglected Rate of fuel consumption, me = 5 kg/ Exhaust velocity, V, - 3500m/s, relative to rocket, leaving at atmospheric pressure Find: (a) Initial acceleration of the rocket. (h) Rocket velocity after 10 s. Solution: Choose a control volume as shown by dashed lines. Because the control volume is accel crating, define inertial coordinate system XY and coordinate system xv attached to the CV. Apply they component of the momentum equation Governing equation Assumptions: Atmospheric pressure acts on all surfaces of the CV; since air resistance is neglected. Fs 0. 2 Gravity is the only body force, g is constant 3 Flow leaving the rocket is uniform, and V, is constant. Under these assumptions the momentum equation reduces to (1) 0 Let us look at the equation term by term: {since ge is constant The mass of the CV will be a function of time because mass is leaving the CV at rate ... To determine Mev as a function of time, we use the conservation of mass equation pe + pv-dio Then 1.pd--L.V.dA=- SAV.,da)== The minus sign indicates that the mass of the CV is decreasing with time. Since the mass of the CV is only a function of time, we can write der To find the mass of the CV at any time. I, we integrate Imamov = ['im, de -- where at 0, Mey - Mandata1,McM
Problem 6. Rocket problem Consider the rocket problem of Example 4.12 (on page 132 in the textbook, 8h edition, or page 103 in the 9h edition): (a) Determine the height at which it consumes all of its fuel, if it's "dry mass” (mass of the rocket after consuming all of its fuel) is 100 kg. (b) Derive the governing differential equations (do not solve!) that describe the height of the rocket, y, at any time after launch, if the rocket experiences a drag force (force of air resistance) opposing its motion that is equal to: F = COPOWA Here, CD is the drag coefficient, and Ar is the "frontal" area of the rocket. Assume an isothermal model for the atmosphere to model the variation in atmospheric air density with altitude, i.e.: ply) = P(O)exp(-2 yo where yo By, and average temperature Toy can be assumed to be at Tw – 250K. Put these equations into forms first- order (in time) for rocket velocity Vx, mass, M, and height, y, (c) This part only: optional and extra credit. Hint: use the starter code provided. Also refer to tutorial videos on how to do this in Matlab Solve the rocket problem, with the modeled air resistance, using the ordinary differential equations solver, ode45, in Matlab. Use a drag coefficient of Co=0.5, and a frontal area Ar=0.2 m? Take the density of air at sea level to be 1.21 kg/ml Plot the height and speed as a function of time, to the point where the rocket has consumed all of its propellant. Compare this to the case of no air resistance.