Motion with variable mass

At time t = 0 a rocket is launched. The rocket has initial mass M, of which mass , 0 1, is fuel. The rocket is launched vertically upwards, from rest, from the surface of the Earth. The rocket burns fuel and the burnt fuel is ejected vertically downwards with constant speed U relative to the rocket. At time t, the rocket has mass m and velocity v. Ignoring air resistance and any variation in g,

show, from first principles, that until all the fuel is used,

The rocket accelerates vertically upwards with constant acceleration g.

Find, in terms of M, U and , an expression for the kinetic energy of the rocket at the instant when all of the fuel has been used.

A small object P, of mass m₀, is projected vertically upwards from the ground with speed U. As P moves upwards it picks up droplets of moisture from the atmosphere. The droplets are at rest immediately before they are picked up. In a model of the motion, P is modelled as a particle, air resistance is assumed to be negligible and the acceleration due to gravity is assumed to have the constant value of g. When P is at a height x above the ground, the combined mass of P and the moisture is m₀(1 + kx), where k is a constant, and the speed of P is v.

Show that, while P is moving upwards

The general solution of this differential equation is given by v² = − 1 + kx, where A is an arbitrary constant.

find, in terms of h, the height of P above the ground when P first comes to rest.

A firework rocket, excluding its fuel, has mass m₀ kg. The rocket moves vertically upwards by ejecting burnt fuel vertically downwards with constant speed u m s^–1, u > 24.5, relative to the rocket. The rocket starts from rest on the ground at time t = 0. At time t seconds, t 2, the speed of the rocket is v m s^–1 and the mass of the rocket including its fuel is m₀(5 – 2t) kg. It is assumed that air resistance is negligible and the acceleration due to gravity is constant.

Show that, for t 2

Find the speed of the rocket at the instant when all of its fuel has been burnt.

A particle P, whose initial mass is m₀, is projected vertically upwards from the ground at time t = 0 with speed , where k is a constant. As the particle moves upwards it gains mass by picking up small droplets of moisture from the atmosphere. The droplets are at rest before they are picked up. At time t the speed of P is v and its mass has increased to m₀e^kt. Assuming that, during the motion, the acceleration due to gravity is constant,

show that, while P is moving upwards,

find, in terms of m₀, the mass of P when it reaches its greatest height above the ground.

A raindrop falls vertically under gravity through a stationary cloud. At time t = 0, the raindrop is at rest and has mass m₀. As the raindrop falls, water condenses onto it from the cloud so that the mass of the raindrop increases at a constant rate c. At time t, the mass of the raindrop is m and the speed of the raindrop is v. The resistance to the motion of the raindrop has magnitude mkv, where k is a constant. Show that

A spacecraft is moving in a straight line in deep space. The spacecraft moves by ejecting burnt fuel backwards at a constant speed of 2000 m s^–1 relative to the spacecraft. The burnt fuel is ejected at a constant rate of c kg s^–1. At time t seconds the total mass of the spacecraft, including fuel, is m kg and the speed of the spacecraft is v m s^–1.

Show that, while the spacecraft is ejecting burnt fuel,

At time t = 0, the mass of the spacecraft is M₀ kg and the speed of the spacecraft is 2000 m s^–1. When t = 50, the spacecraft is still ejecting burnt fuel and its speed is 6000 m s^–1.

upwards from rest. The rocket burns fuel at a rate of 15 kg s! and the burnt fuel is ejected vertically downwards with a speed of 1000 m relative to the rocket. At time ¢ seconds

Show that

A rocket propels itself by its engine ejecting burnt fuel. Initially the rocket has total mass M, of which a mass kM, k 1, is fuel. The rocket is at rest when its engine is started. The burnt fuel is ejected with constant speed c, relative to the rocket, in a direction opposite to that of the rocket’s motion. Assuming that there are no external forces, find the speed of the rocket when all its fuel has been burnt.

A raindrop falls vertically under gravity through a cloud. In a model of the motion the raindrop is assumed to be spherical at all times and the cloud is assumed to consist of stationary water particles. At time t 0, the raindrop is at rest and has radius a. As the raindrop falls, water particles from the cloud condense onto it and the radius of the raindrop is assumed to increase at a constant rate λ. A time t the speed of the raindrop is v.

Show that

Find the speed of the raindrop when its radius is 3 a.

A spaceship is moving in a straight line in deep space and needs to increase its speed. This is done by ejecting fuel backwards from the spaceship at a constant speed c relative to the spaceship. When the speed of the spaceship is v, its mass is m.

Show that, while the spaceship is ejecting fuel,

The initial mass of the spaceship is m₀ and at time t the mass of the spaceship is given by m = m₀ (1 – kt), where k is a positive constant.

Find the acceleration of the spaceship at time t.

At time t = 0 a rocket is launched from rest vertically upwards. The rocket propels itself upwards by expelling burnt fuel vertically downwards with constant speed U m s –1 relative to the rocket. The initial mass of the rocket is M₀ kg. At time t seconds, where t < 2, its mass is M (1– 0 kg, and it is moving upwards with speed t)v m s–1.

Show that

Hence show that U > 19.6

Find, in terms of U, the speed of the rocket one second after its launch.