The first cosmic speed is a speed at which an object can become the artificial Earth satellite. Its size (about 8 km/s) is known by any school student. At the same time many forget that speed has not only the size, but also the direction.
Instruction
1. If to disperse some body to even very high speed in the vertical direction, then having spent stocks of kinetic energy, it all the same will return back. The reason – absence in this direction of any forces except terrestrial gravitation. Means there has to be a force counteracting gravity. Such force takes place if the body participates in rotary motion concerning Earth (more precisely than its center). At such movement there is a centripetal acceleration creating force directed opposite to inclination.
2. Now you have the right to draw a conclusion that the considered speed is directed on a tangent to a circle on which the body moves. It is called the linear speed of rotary motion. If m body weight, then the corresponding centrifugal force of Fц=m(v^2/L), where L – distance from the center of Earth to a body. If to use height of h of removal of a body from a surface, then L=R+h (R is Earth radius).
3. The gravitational force of an attraction of two masses γ (Mm/L^2)=γ (Mm / (R+h) ^2), where γ – a gravitational constant. Equate it to centrifugal force (see fig. 1). Receive γ (Mm / (R+h) ^2) = m (v^2 / (R+h)). Carry out reductions and find v. v= √ (γM / (R+h). Now consider that h is negligible in comparison with R or R+h≈R. Consider also that the acceleration of gravity of g=γМ/R^2. Then v= √ (gR). R=6357*(10^3) of m, g=9.81 м/с^2 and then km/s v≈7.91.
4. Together the question of how to be with h height cannot but arise that, at end of the solution of a task on the first cosmic speed? Really, this size did not participate in final settlement. The fact is that even those heights at which near-earth spacecrafts fly are too small in comparison with earth radius. And masses them is also insignificant in comparison with the mass of our planet. Nevertheless, to h there is the main requirement. It is more than thickness of the atmosphere. In its dense layers the bodies with such speeds just burn down.