Newton's three Laws of Motion

Rocket Equation

**Newton's three Laws of Motion**

Isaac Newton's Laws of Motion were first published in his *Philosophiae
Naturalis Principia Mathematica* (1687).

**Newton's first law**: law of inertia

*Lex I: Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter
in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.*

*** Unless acted upon by an unbalanced force, an object will maintain
a constant velocity**.

*Note*: Velocity is a vector, therefore a constant velocity is defined
as a constant speed on an unchanging direction (i.e. a linear path).

An object may be acted upon by many forces and maintain a constant velocity so long as these forces are balanced. For example, a rock resting upon the Earth keeps a constant velocity (in this case, zero) because the downward force of its weight balances out the upward force (called the normal force) which the Earth exerts upwardly on the rock. Only unbalanced forces induce acceleration, or a change in the velocity or an object. If you push someone, he or she will accelerate in the direction of the unbalanced force which you have provided (called the applied force). Likewise if you roll a ball along the floor, the unbalanced force of friction will decelerate the ball from some positive velocity to rest.

Before Galileo, people agreed with Aristotle that a body's natural state was at rest, and that movement needed a cause. This is understandable, since in everyday experience, moving objects eventually stop because of friction (except for celestial objects, which were deemed perfect). Moving from Aristotle's "A body's natural state is at rest" to Galileo's discovery was one of the most profound and important discoveries in physics.

There are no true examples of the law, as friction is usually present, and even in space gravity acts upon an object, but it serves as a basic axiom for Newton's mathematical model from which one could derive the motions of bodies from elementary causes: forces. Another way to put it is ,"An object in motion tends to stay in motion, an object at rest tends to stay at rest until a force acts upon it"

**Newton's second law**

*Lex II: Mutationem motus proportionalem esse vi motrici impressae et fieri
secundum lineam rectam qua vis illa imprimitur.*

*** The rate of change of momentum of a body is equal to the resultant
force acting on the body and is in ****the same direction.**

When the mass m of the object is constant, the second law can be written:

F = ma

i.e. Force = mass x acceleration

For example, if a bowstring exerts a constant force of 100 newtons on an arrow having a mass of 0.10 kg, then the arrow's acceleration will be 1000 m/s2 until it leaves the bow (after which the arrow will stop speeding up).

In these equations, **F** is the net force, i.e., the sum of all
the forces acting on the object. When the forces on the object all act along
the same line, they can be added as positive and negative numbers, depending
on their direction. When they do not all act along the same line, the total
must be found by vector addition.

The quantity m, or mass, is a characteristic of the object. The greater the total force acting on an object, the greater the change in its acceleration will be. This equation, therefore, indirectly defines the concept of mass. In the equation, F = ma, a is directly measurable but F is not. The second law only has meaning if we are able to assert, in advance, the value of F. Rules for calculating force include Newton's law of universal gravitation, Coulomb's law, and other principles.

**Newton's third law**: law of reciprocal actions

*Lex III: Actioni contrariam semper et aequalem esse reactionem: sive corporum
duorum actiones is se mutuo semper esse aequales et in partes contrarias dirigi.*

*** All forces occur in pairs, and these two forces are equal in magnitude
and opposite in direction.**

If a basketball hits the ground, the basketball's force on the Earth is the same as Earth's force on the basketball. However, due to the ball's much smaller mass, Newton's second law predicts that its acceleration will be much greater. Not only do planets accelerate toward stars; but, stars accelerate toward planets.

The two forces in Newton's third law are of the same type, e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road.

**Tsiolkovsky rocket equation**

The *Tsiolkovsky rocket equation*, or *ideal rocket equation*, or
simply *rocket equation*, relates the maximum change of speed of a rocket
- which occurs when there are no external forces acting on the system - to the
effective exhaust velocity and the initial and final mass of a rocket (or other
reaction engine).

For any series of thrusts:

where:

- Δ
*v*is the maximum change of speed of the rocket. *v*_{e}is the effective exhaust velocity.*m*_{0}is the initial total mass, including propellant.*m*_{1}is the final total mass.

The units used for mass or velocity do not matter as long as they are consistent.