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.
Suppose = 200
kg, R = 2000
N, λ = 1 kg/s, = 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.
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 λ. 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?