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Mathematics

Newton's Laws of Motion: An Intuitive Guide

A brief exploration of how objects move, why they change speed, and the hidden force-pairs that define interactions in our universe.

Frictionless Ice
∑F⃗=0  ⟹  v⃗=constant\sum \vec{F} = 0 \implies \vec{v} = \text{constant}
Why does an object keep moving on its own? Imagine a puck on perfectly smooth ice; without friction, it has no reason to speed up, slow down, or turn. It simply persists in its state of motion until an external force interferes.
Step-by-step solver
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(a) First Law

State the law: an object at rest or constant velocity remains that way unless a net external force acts on it. Inertia describes this inherent resistance to change in motion.

∑F⃗=0  ⟹  v⃗=constant\sum \vec{F} = 0 \implies \vec{v} = \text{constant}
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(b) Second Law

Relate force to mass and acceleration by deriving it from momentum. Since mass is constant, the derivative of momentum becomes mass times the derivative of velocity (acceleration).

F⃗=d(mv⃗)dt=ma⃗\vec{F} = \frac{d(m\vec{v})}{dt} = m\vec{a}
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(c) Third Law

Explain that forces are always interactions between two objects. The force object A exerts on B is equal in magnitude and opposite in direction to the force B exerts on A.

F⃗A→B=−F⃗B→A\vec{F}_{A\to B} = -\vec{F}_{B\to A}

Original question

Newton's laws animated

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