Movement of the system with variable mass

In nature and technology often we meet such a body or system of bodies,

Mass of which changes during movement, such as different classes of reactive missiles,

shells, ice floe, which floats and melts, the spindle, from which the thread is wound, and

etc. System, mass of which is constantly changing, as a result of accession or separation from her some bodies, is called system with variable mass.

Consider a body of mass M (Fig. 2.5).

Suppose that body M in the moment of time t has the velocity and body dM – velocity . Impulse of the system of two bodies before collision in the moment of time t

As a result of inelastic collision the mass M increases and in the moment t+dt is equal M+dM and velocity is . Impulse of the system in the moment of time t+dt

Thus, the change of momentum is written in the form:


Ignoring value dmdv of highest order of smallness dmdv formula (2.23) is written down as:

According to the second Newton’s law:

Where is sum of external force, that affect the system.

Thus, from (2.34) and (2.35):


From formula of the addition of the velocity of classic mechanic follows that , where the velocity of the body with mass dM relatively to the body M.

Then the formula (2.36) takes the form:

Relation (2.37) is the equation of motion of a system with variable mass, it is called variable mass system. The first item on the right side of the equation (2.37) describes the total external force that acts on the system (in the case of the rocket advisable to include gravity, air resistance force.) The second item on the right side of (2.37) has the dimension of the force, and therefore it can be interpreted as the force that occurs as a result of changes body mass M. It is called reactive power.

Consider the solution of variable mass system (2.37) regarding the movement of the rocket, which is not acted upon by external forces ( ). Take as relative velocity, that is constant in magnitude and direction, of particles, that separates. Moreover the opposite of the initial velocity vector (Fig. 2.6).

Formula (2.37) get the form:

Project the equation (2.38) on the vertical axis y:

Integrate (2.39)

where M - mass of the missile at the time when the velocity becomes a value v,

- initial mass of the rocket, which contains a lot of fuel and mass of the body of

rocket with all it’s equipment.

Formula (2.41) was received by Tsiolkovsky in 1903, and it was named in his

honor. From this it follows that the velocity v, which in the absence of the missile acquires

external force is directly proportional to (relative velocity of the particles that

separate), and the natural logarithm of the ratio of primary and the final mass of the rocket.

The velocity v does not depend on what the law is changed rocket weight: it is important to know only the initial and final values of the mass.

take in the formula (2.41) , ie, the initial velocity of the rocket equal to zero. Suppose, for example, that you must give first rocket escape velocity, that is, such that it has started to move around the Earth circle. This speed is v=8km/s Also assume that the leak rate of gas jet is , then from 2.41 it follows that = 2980, ie,

the initial mass of the rocket exceeds the final in 2980 times. Almost all the entire mass

of rocket should belong to the mass of fuel. Increasing , for example, to 4km/s we get = 7.39. That means, if we increase the speed of the gas leak, it will be possible to increase the useful weight of the rocket.

The idea of producing cosmic velocities with rockets for the first time was formulated by Tsiolkovsky. He suggested the use of so-called multistage rocket. First, the engine is running the first stage. When fuel this degree is completely burned, it is separated from the main body and this moment will start to work the engines of the second stage, etc.