CLASSICAL MECHANICS
Specificity of Newton's laws for objects with variable mass

Interesting facts about classical mechanics and rocket motion

What is escape velocity?

If you throw an object straight up, it will rise until the the negative acceleration of gravity stops it, then returns it to Earth. Gravity's force diminishes as distance from the center of the Earth increases, however. So if you can throw the object with enough initial upward velocity so that gravity's decreasing force can never quite slow it to a complete stop, its decreasing velocity can always be just high enough to overcome gravity's pull. The initial velocity needed to achieve that condition is called escape velocity.

From the surface of the Earth, escape velocity (ignoring air friction) is about 7 miles per second, or 25,000 miles per hour. Given that initial speed, an object needs no additional force applied to completely escape Earth's gravity.

What is the difference between the inertial and gravitational mass?

The only difference that we can find between inertial and gravitational mass that we can find is the method.

Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done with some sort of balance scale. The beauty of this method is that no matter where, or what planet, you are, the masses will always balance out because the gravitational acceleration on each object will be the same. This does break down near supermassive objects such as black holes and neutron stars due to the high gradient of the gravitational field around such objects.

Inertial mass is found by applying a known force to an unknown mass, measuring the acceleration, and applying Newton's Second Law, m = F/a. This gives as accurate a value for mass as the accuracy of your measurements. When the astronauts need to be weighed in outer space, they actually find their inertial mass in a special chair.

The interesting thing is that, physically, no difference has been found between gravitational and inertial mass. Many experiments have been performed to check the values and the experiments always agree to within the margin of error for the experiment. Einstein used the fact that gravitational and inertial mass were equal to begin his Theory of General Relativity in which he postulated that gravitational mass was the same as inertial mass and that the acceleration of gravity is a result of a 'valley' or slope in the space-time continuum that masses 'fell down' much as pennies spiral around a hole in the common donation toy at your favorite chain store.

If space is a vacuum, then how do space capsule's retrorockets steer?

The fact that space is a vacuum nicely isolates this problem as a demonstration of Newton's third Law, commonly phrased as 'every action has an equal and opposite reaction.'

It is not necessary for the rocket exhaust to push against anything EXCEPT THE SHIP ITSELF. You see, when the combustion of fuel takes place inside the rocket (think of a long vertical cylinder with the 'bottom' open) the exhaust gasses produced expand quickly in all directions. The molecules slam into anything in their path exerting a small force each. Thus some molecules push against the 'right' side and some push against the 'left' side, and all these sideways forces cancel each other out. Some molecules slam against the 'top' of the cylinder, but since there is no bottom of the cylinder, there is no force to cancel this out! Therefore the net force will be in the 'up' direction.

Another way to think about the situation is as a conservation of momentum problem. Any isolated system will conserve total momentum. Thus if you imagine a stationary ship, plus the fuel and oxygen molecules on the ship, p=0. If a few trillion atoms of exhaust are shot in one direction into the vacuum of space, the rest of the ship must move in the OPPOSITE direction with the same momentum (mass x velocity)

You can simulate this situation by throwing some bean bags (or bowling balls, or any other objects you have lying around) in one direction as you sit in a chair with wheels or stand on roller blades or ice skates. Whatever direction you push on the object, the object will push back on your hand with the same force, but in the opposite direction. Therefore you and the object you throw will move in opposite directions.

Is the rotation of the earth considered the initial kinetic energy for a rocket before take-off?

Yes, a rocket sitting on its launching pad on Earth has an eastward velocity due to the rotation of the Earth. That is why almost all rockets take an easterly trajectory after a vertical launch. (Exceptions would be satellites designed to cover the largest portion of the Earth's surface, placed in a polar orbit to allow the Earth to rotate beneath them.)

The maximum eastern velocity available is at the equator, where a point on the surface travels about 25,000 miles in 24 hours, or over 1000 miles per hour. As you move north or south of the equator, the circle traveled each day, and therefore the speed, becomes smaller until it becomes zero at the poles. This was a prime factor in locating Cape Canaveral as far south, in Florida, as it was. The velocity already imparted by Earth's rotation is that much less needed by the rocket's own fuel to attain the necessary orbital speed.

Interplanetary vehicles travelling to the outer planets can also take advantage of the Earth's orbital velocity around the Sun (about 18 miles per second), while those travelling to the inner planets must reduce the orbital speed imparted by the Earth to 'fall' toward a closer orbit.