The theory of rocket motion developed in Exercise 1, no longer applies in the relativistic...

The theory of rocket motion developed in Exercise 1, no longer applies in the relativistic region, in part because there is no longer conservation of mass. Instead, all the conservation laws are combined into the conservation of the world momentum; the change in each component of the rocket’s world momentum in an infinitesimal time dt must be matched by the value of the same component of pv for the gases ejected by the rocket in that time interval. Show that if there are no external forces acting on the rocket, the differential equation for its velocity as a function of the mass is

where a is the constant velocity of the exhaust gases relative to the rocket. Verify that the solution can be put in the form

m0 being the initial mass of the rocket. Since mass is not conserved, what happens to the mass that is lost?

Exercise 1

Rockets are propelled by the momentum reaction of the exhaust gases expelled from the tail. Since these gases arise from the reaction of the fuels carried in the rocket, the mass of the rocket is not constant, but decreases as the fuel is expended. Show that the equation of motion for a rocket projected vertically upward in a uniform gravitational field, neglecting atmospheric friction, is

where m is the mass of the rocket and v′ is the velocity of the escaping gases relative to the rocket. Integrate this equation to obtain v as a function of m, assuming a constant time rate of loss of mass. Show, for a rocket stalling initially from rest, with v′ equal to 2.1 km/s and a mass loss per second equal to 1/60th of the initial mass, that in order to reach the escape velocity the ratio of the weight of the fuel to the weight of the empty rocket must be almost 300!