Understand that mass is the quantitative measure of an object’s inertia – the property that resists any change in its state of motion.
| Quantity | Scalar / Vector | SI Unit | Typical Uncertainty |
|---|---|---|---|
| Displacement, velocity, acceleration, force, momentum | Vector | m, m s⁻¹, m s⁻², N, kg m s⁻¹ | ± 0.5 % (instrument) + ± last‑digit (reading) |
| Distance, speed, mass, time, kinetic energy | Scalar | m, m s⁻¹, kg, s, J | ± 0.5 % (instrument) + ± last‑digit (reading) |
An object remains at rest or moves with constant velocity unless acted upon by a net external force. The resistance to any change in motion is called inertia, and the magnitude of inertia is quantified by the object's mass (m).
The fundamental statement is the momentum principle:
\[
\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}
\qquad\text{where}\qquad
\mathbf{p}=m\mathbf{v}
\]
For a body of constant mass:
\[
\mathbf{F}_{\text{net}} = \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt}=m\mathbf{a}
\]
Thus the familiar form \(\mathbf{F}_{\text{net}} = m\mathbf{a}\) is a special case of the more general momentum equation.
For every interaction there is an equal and opposite pair of forces:
\[
\mathbf{F}{AB} = -\mathbf{F}{BA}
\]
These forces act on different bodies, so they never cancel when analysing the motion of a single object.
Two common approximations:
\[
\mathbf{F}_{\text{drag}} = -k\mathbf{v}\qquad\text{(linear drag, low speeds)}
\]
\[
\mathbf{F}_{\text{drag}} = -k v^{2}\,\hat{\mathbf{v}}\qquad\text{(quadratic drag, high speeds)}
\]
For a falling object of mass \(m\) under gravity \(mg\) with linear drag, the terminal velocity is obtained when the net force is zero:
\[
mg - kv{t}=0\;\;\Longrightarrow\;\;v{t}= \frac{mg}{k}
\]
Example: A sky‑diver (mass 80 kg) with a linear drag constant \(k=160\;\text{N s m}^{-1}\) reaches a terminal speed \(v_{t}= \frac{80\times9.8}{160}=4.9\;\text{m s}^{-1}\).
\[
\mathbf{p}=m\mathbf{v}
\]
Because mass appears directly, a larger mass gives a larger momentum for the same speed, and therefore a greater resistance to a change in motion.
\[
\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}
\]
\[
\mathbf{J} = \int{t{1}}^{t_{2}}\mathbf{F}\,dt = \Delta\mathbf{p}
\]
Impulse \(\mathbf{J}\) is the area under a force‑time graph and equals the change in momentum.
\[
\sum{\text{system}}\mathbf{p}{\text{initial}} = \sum{\text{system}}\mathbf{p}{\text{final}}
\]
\[
e = \frac{v{2}'-v{1}'}{v{1}-v{2}}\quad(0\le e\le1)
\]
where \(e=1\) for a perfectly elastic impact and \(e=0\) for a perfectly inelastic impact.
Momentum conservation must be applied separately to each component:
\[
\sum p{x,\text{initial}} = \sum p{x,\text{final}},\qquad
\sum p{y,\text{initial}} = \sum p{y,\text{final}}
\]
This allows analysis of glancing blows, billiard‑ball problems, and explosions.
Two gliders on a frictionless air‑track:
\[
m{1}v{1}+m{2}v{2}=m{1}v{1}'+m{2}v{2}'
\]
\[
\tfrac12 m{1}v{1}^{2}+\tfrac12 m{2}v{2}^{2}
=\tfrac12 m{1}{v{1}'}^{2}+\tfrac12 m{2}{v{2}'}^{2}
\]
\[
v{1}'=-2.0\;\text{m s}^{-1},\qquad v{2}'=+2.0\;\text{m s}^{-1}
\]
Glider A rebounds, Glider B moves forward with the same speed.
Same masses, but the gliders stick together.
Using momentum conservation:
\[
p{i}=m{\text{tot}}v{f}\;\Longrightarrow\;v{f}= \frac{2.0}{2.0}=1.0\;\text{m s}^{-1}
\]
Kinetic energy loss:
\[
K{i}= \tfrac12 m{1}v_{1}^{2}=4.0\;\text{J},\qquad
K{f}= \tfrac12 m{\text{tot}}v_{f}^{2}=1.0\;\text{J}
\]
The missing 3 J is transformed into internal energy (sound, deformation).
Ball 1 (\(m\)) moves at \(5.0\;\text{m s}^{-1}\) along the +x‑axis, strikes identical stationary ball 2. After impact, ball 1 moves at \(3.0\;\text{m s}^{-1}\) at \(30^{\circ}\) above the x‑axis. Find the speed of ball 2.
\[
mv{1}=m v{1}'\cos30^{\circ}+m v_{2}'\cos\theta
\]
\[
0=m v{1}'\sin30^{\circ}+m v{2}'\sin\theta
\]
| Law / Principle | Mathematical Form | Physical Meaning | Role of Mass |
|---|---|---|---|
| Newton’s First Law | \(\mathbf{F}_{\text{net}}=0\;\Rightarrow\;\mathbf{v}= \text{constant}\) | Objects maintain their state of motion unless a net external force acts. | Mass quantifies the inertia that resists any change in motion. |
| Newton’s Second Law (derived) | \(\displaystyle\mathbf{F}_{\text{net}}=\frac{d\mathbf{p}}{dt}=m\mathbf{a}\) | Force produces an acceleration proportional to the mass. | Mass is the proportionality constant linking force and acceleration. |
| Newton’s Third Law | \(\mathbf{F}{AB}=-\mathbf{F}{BA}\) | Forces occur in equal‑and‑opposite pairs on interacting bodies. | Each body’s mass determines the acceleration produced by the pair of forces. |
| Momentum Definition | \(\mathbf{p}=m\mathbf{v}\) | Momentum combines mass and velocity into a conserved vector quantity. | Mass directly scales momentum, increasing resistance to change. |
| Impulse–Momentum Theorem | \(\displaystyle\mathbf{J}= \int\mathbf{F}\,dt = \Delta\mathbf{p}\) | Impulse (area under a force‑time graph) equals the change in momentum. | Mass appears in \(\Delta\mathbf{p}=m\Delta\mathbf{v}\). |
| Conservation of Linear Momentum | \(\displaystyle\sum\mathbf{p}{\text{initial}} = \sum\mathbf{p}{\text{final}}\) (isolated system) | Total momentum of a closed system remains constant. | Distribution of mass among the objects dictates how momentum is shared after interaction. |
| Linear Drag (non‑uniform motion) | \(\mathbf{F}{\text{drag}}=-k\mathbf{v}\) or \(\mathbf{F}{\text{drag}}=-k v^{2}\hat{\mathbf{v}}\) | Resistive force proportional to speed (low‑speed) or speed squared (high‑speed). | Mass determines the terminal speed \(v_{t}=mg/k\) (linear model). |
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