Sir Isaac Newton’s Laws of Motion

First published in 1686.

Amazon: Philosophiae Naturalis Principia Mathematica
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First Law of Motion

Also known as the Law of Inertia

An object at rest remains at rest unless acted on by an unbalanced force. An object in motion continues with constant velocity in a straight line unless acted upon by an unbalanced force.

\[ \sum \mathbf{F} = 0 \] \[ \frac{d\mathbf{v}}{dt} = 0 \]

In every material universe, the motion of a particle in an inertial frame Φ is determined by forces whose total vanishes when and only when the velocity of the particle is constant in Φ.

Newton’s first and second laws apply only in inertial reference frames. Any frame moving at constant velocity relative to an inertial frame is also inertial (Galilean invariance).

Second Law of Motion

The second law states that the rate of change of momentum of a body is directly proportional to the force applied, and this change in momentum takes place in the direction of the applied force.

Momentum (p) is velocity times mass.

Momentum:

\[ p = mv \]

Newton’s Second Law:

\[ F = \frac{dp}{dt} \]

\[ \mathbf{F} = \frac{d\,(m\mathbf{v})}{dt} \]

\[ \mathbf{F} = \frac{d(m\mathbf{v})}{dt} \] \[ = \frac{m\,d\mathbf{v}}{dt} \] \[ = m\mathbf{a} \text{ , since} \frac{d\mathbf{v}}{dt} = \mathbf{a} \]

This is commonly remembered as:

\[ F = ma \]

Impulse

Impulse (J) occurs when a force acts over a time interval.

so Impulse = Force x Time

\[ J = Ft \]

Impulse equals the change in momentum:

\[ \mathbf{J} = \int_{\Delta t} \mathbf{F}\,dt \]

\[ J = \Delta p \]

Third Law of Motion

All forces between two objects exist in equal magnitude and opposite direction. If object A exerts a force \(F_A\) on object B, then B exerts a force \(F_B\) on A such that:

\[ F_A = -F_B \]

Here, \(F_A\) is the action and \(F_B\) is the reaction.