Question

A rocket starts at rest at t=0 with initial mass consisting of a payload of mass M together with fuel of mass 9M. The fuel is burnt at a constant rate λ with the mass being emitted backwards at speed u relative to the rocket. Let u(t) be the rocket's speed at time t and consider the situation of constant gravitational force (of magnitude g per unit mass) but negligible air resistance. Find the differential equation satisfied by v. dw/3 = 2. The rocket will only leave the ground if the rate of fuel burning λ is greater than a critical value λ. Determine an expression for λi in terms of M,g, and u. . Assuming the rocket is able to leave the ground, what is its final velocity vf at the time when all the fuel is burnt? vf =

Answer 1

The differential equation satisfied by the rocket's velocity, v, can be derived using the principle of conservation of momentum. The critical value of lambda can be determined by setting the weight-to-thrust ratio equal to 1. When all the fuel is burnt, the rocket's final velocity is equal to the exhaust gas velocity.

Step-by-step explanation:

The differential equation satisfied by the rocket's velocity, v, can be derived using the principle of conservation of momentum. The momentum of the rocket and its exhaust gases is given by:

m(t) * v(t) + dm(t) * (v(t) + u(t)) = (m(t) - dm(t)) * (v(t) + dv(t))

By simplifying and taking the limit as dm(t) approaches 0, we obtain the differential equation:

dv(t)/dt = -u(t) * dm(t)/dt / (m(t) - dm(t)) - g * dm(t)/dt / (m(t) - dm(t))

where v(t) is the rocket's velocity, u(t) is the exhaust gas velocity, m(t) is the rocket's mass at time t, dm(t)/dt is the rate of mass loss, and g is the acceleration due to gravity.

The critical value of lambda, denoted as lambda_i, can be determined by setting the weight-to-thrust ratio equal to 1. The weight of the rocket is given by M * g, and the thrust is lambda * u. Therefore, lambda_i = M * g / u.

When all the fuel is burnt, the rocket's final velocity, vf, can be found by solving the equation v(t) + u(t) = vf. Since u(t) is constant, vf = v(t) + u(t) = 0 + u(t) = u(t).