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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 As a result of inelastic collision the mass M increases and in the moment t+dt is equal M+dM and velocity is Thus, the change of momentum is written in the form:
Ignoring value dmdv of highest order of smallness dmdv According to the second Newton’s law:
Where Thus, from (2.34) and (2.35):
Or
From formula of the addition of the velocity of classic mechanic 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) Consider the solution of variable mass system (2.37) regarding the movement of the rocket, which is not acted upon by external forces
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,
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 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) 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 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.
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