Consider a small single stage rocket launched vertically. Let m(t) denote the to
ID: 2942165 • Letter: C
Question
Consider a small single stage rocket launched vertically. Let m(t) denote the total mass of the rocket at time t (mass of rocket is the sum of three masses: constant mass of payload, constant mass of the vehicle, and the variable amount of fuel). It is assumed that the positive direction is upward, air resistance (drag) is proportional to the instantaneous velocity v of the rocket (constant of proportionality is k) and R is the constant upward thrust or force generated by the propulsion system then,Find the DE that defines the velocity v(t) of the rocket (assuming g as the gravitational constant)
Suppose that m(t)= m_p+m_v+m_f (t), m_p is the constant mass of the payload, m_v is the constant mass of the vehicle, and m_f (t) is the variable mass of the fuel.
Show that the rate at which the total mass of the rocket changes is the same as the rate at which the mass of the fuel changes.
If the rocket consumes its fuel at the constant rate of ? (??>0), find m(t). Then rewrite the differential equation in terms of ??and the initial total mass? m(0)= m_0?
Under the assumption in part (b), show that the burnout time t_b>0 of the rocket, or the time at which all the fuel is consumed, is t_b=m_f (0)/?, where m_f (0) is the initial mass of the fuel
Explanation / Answer
x = 3