A small single-stage rocket is launched vertically upward as shown in the figure. Once launched, let us assume that the rocket consumes its fuel at a constant rate, causing the total mass m(t) to vary with time t. Here, the total mass m(t) is the sum of three different masses: the constant mass of the payload, the constant mass of the vehicle, and the variable mass of the fuel. Assume also that the positive direction is upward, the atmospheric drag (air resistance) is proportional to the velocity v(t) of the rocket where k is the proportionality constant, and, R is the upward thrust generated by the propulsion system (Links to an external site.)Links to an external site.. Now, answer the following questions:

1.Write an equation of motion for the rocket. Hint: Pay particular attention to the fact that m(t) is NOT a constant quantity. So you must be picky about choosing an appropriate version of Newtons 2nd law (Links to an external site.)Links to an external site..

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

3.Suppose m 0 = 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 velocity v(t) at any time t.

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