A small single-stage rocket is launched verticallyupwardas shown in the figure. Once launched, let us assumethatthe rocket consumes its fuel at a constant rate, causingthe total massm(t)tovary with timet.Here, the total massm(t)is the sum of three different masses: the constant mass of the payload, the constant mass of thevehicle, and the variable mass of the fuel.Assumealsothat the positive direction is upward, the atmospheric drag (air resistance) is proportional to thevelocityv(t)of the rocketwherekisthe proportionality constant,and,Ris the upward thrust generated by thepropulsion system(Links to an external site.)

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Now, answer the following questions:

1.discuss an equation of motionfor the rocket.Hint: Pay particular attention to the fact thatm(t)is NOT a constant quantity. So you must be picky about choosing an appropriate version ofNewton's 2ndlaw

2.Letm(0)=m0 be the initial total mass of the rocket, andbe the constant rate of fuel consumption.Now rewrite the differential equation obtained in part (a) in terms ofm0and.

3. Supposem0 = 200 kg,R= 2000 N,= 1 kg/s,g= 9.8 m/s², k= 3 kg/s, and the rocket is launched from rest on the ground. Find its velocityv(t)at any timet.

4.The burnout time(t_b)of the rocket is the time at which all the fuel is consumed.The value oft_bis the ratio of the initial amount of fuel to thefuel consumptionrate. Assumingthe initial amount of fuel as 60 kg, find the value of tb.How fast will the rocket be moving at the time of burnout?