*Notes to a video lecture on http://www.unizor.com*

__Rocket Calculation 1__

Let's use the rocket equation

Δ

**·**

*V = −v*_{e}*ln*[

**]**

*m(t*_{beg})/m(t_{end})to calculate how much propellant must be taken by a rocket to reach an orbit.

Here

**is effective exhaust speed,**

*v*_{e}**- mass of a rocket in the beginning of a time period during which a rocket's engine is working,**

*m(t*_{beg})**- mass of a rocket at the end of this period of acceleration or deceleration.**

*m(t*_{end})Recall that the minus sign in this equation signifies the vector

character of the movement: positive direction of the exhausted

propellant (that is, the same as the movement of the rocket) causes

negative increment in rocket's speed - deceleration, while the negative

direction of exhausted propellant (that is, opposite to the movement of a

rocket) causes increase in rocket's speed - acceleration.

Contemporary rocket engine can have a very high effective exhaust

velocity. The speed of about 4km/sec is mentioned in a few sources we

are familiar with. So, we can assume that

*v*_{e}=4000m/sec.An international Space Station's speed is about 7.8km/sec, as was calculated in one of the previous lectures on gravity.

Assuming that the initial speed of a rocket is zero, the increment of speed of a rocket must be

Δ

*V = 7800m/sec*From this follows that

*ln*[

**]**

*m(t*_{beg})/m(t_{end})

*=*

= 7800/4000 = 1.95= 7800/4000 = 1.95

Therefore,

*m(t*_{beg})/m(t_{end}) ≅ 7So, the mass of a rocket at start is 7 times greater than its mass at

the end of its acceleration. For example, to launch 1,000 kg of useful

equipment and/or passengers to an International Space Station we will

need 6,000 kg of fuel.

__Rocket Calculation 2__

We still assume that

*v*_{e}=4000m/sec.A rocket that goes far from Earth needs about 11.2km/sec speed to escape Earth gravity.

Assuming that the initial speed of a rocket is zero, the increment of speed of a rocket must be

Δ

*V = 11200m/sec*From this follows that

*ln*[

**]**

*m(t*_{beg})/m(t_{end})

*=*

= 11200/4000 = 2.8= 11200/4000 = 2.8

Therefore,

*m(t*_{beg})/m(t_{end}) ≅ 16So, the mass of a rocket, that is supposed to leave the Earth's gravity,

at start is 16 times greater than its mass at the end of its

acceleration. For example, to launch 1,000 kg of useful equipment and/or

passengers to Mars we will need 15,000 kg of fuel.

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