Air pressure to launch rockets?

An idea I had when I was very young, can we launch rockets just using pressure and no fuel? The answer is: without a big explosion no, we can’t:

The idea of using air pressure to create a positive force capable of launching a rocket had already been put into practice in small homemade “rockets”.

First I will discuss the mathematical model used (sorry the images are in spanish):

Basically the idea consist of a long vertical columm, the top part is vaccummed so the rocket inside moves forward:

In a normal situation, Air Pressure Air Pressure 1 = 2, thus no resultant force. When a pressure difference is applied, a displacement of air 2 -> 1 occurs, thereby moving the object.

– Space 2 is closed by a top closure. Let’s express Newton’s second law:

– Where P1 is the air pressure P 1 and the weight of the load and A is the area of the rocket in air contact 1 and 2.
Applying the ideal gas law:

P1 decrease as the fluid in one go spread, will each 100 meters the pressure is reduced by k, KPa. Volume V anger also decreasing as the rocket rises as the space between the top closure and the rocket will be reduced. For a cylinder looks like this:

– Where x is the vertical position of the rocket, the height h of the cylinder (column) r the radius.
Now we have the function of the acceleration of the rocket as a function of x, with a little calculation llgearemos to the function of speed:

Integrating parts:

Isolate v (x):

Isn’t that beatiful? Now anyone can play around with the values in this equation:

To get the speed of the rocket at the top part which is what we want to calculate, take into consideration that to escape earth gravity you need to move at around 12.2 km/s (air friction calculated).