Core High-Yield Fact Ledger
- Kinematics describes motion without considering causes; dynamics evaluates motion along with acting forces.
- Rest and motion are relative to the chosen observer; no body in the universe is in absolute rest or motion.
- Distance is a scalar, strictly positive ($S > 0$), and monotonically increases; displacement is a vector from start to finish that can be positive, negative, or zero.
- The ratio of distance to displacement magnitude is always $\frac{S}{|\vec{d}|} \ge 1$; equality holds strictly for unidirectional straight-line motion.
- For a semicircular track of radius $r$, distance is $\pi r$ and displacement is $2r$, producing the ratio $\frac{\pi}{2} \approx 1.57$.
- Average velocity is net displacement over total time ($\vec{v}_{\text{avg}} = \frac{\Delta \vec{d}}{\Delta t}$); uniform velocity means instantaneous velocity equals average velocity.
- Relative speed: Opposite directions add ($v_1 + v_2$); identical directions subtract ($|v_1 - v_2|$).
- Acceleration points in the direction of the change in velocity ($\Delta \vec{v}$), not necessarily along velocity ($\vec{v}$).
- Speeding up means $\vec{a} \parallel \vec{v}$ ($\theta = 0^\circ$); slowing down means $\vec{a}$ is antiparallel to $\vec{v}$ ($\theta = 180^\circ$); circular turning means $\vec{a} \perp \vec{v}$ ($\theta = 90^\circ$).
- At the peak of vertical throw: instantaneous velocity is 0 m/s, but acceleration is 9.8 m/s^2 downward.
- Displacement-time graph slope equals instantaneous velocity; distance-time slope can never be negative.
- Velocity-time graph slope equals acceleration; area under velocity-time graph equals displacement or distance.
- Acceleration-time graph area equals net change in velocity ($\Delta v = v_f - v_i$).
- Equations of rectilinear motion apply strictly to uniform, constant acceleration only.
- Galileo odd number law for bodies dropped from rest: distances fallen in consecutive equal time intervals follow the ratio 1 : 3 : 5 : 7.
- For free fall from rest under gravity ($g = 10\text{ m/s}^2$): $t = 1\text{ s} \implies h = 5\text{ m}$; $t = 2\text{ s} \implies h = 20\text{ m}$; $t = 3\text{ s} \implies h = 45\text{ m}$.
- Newton's First Law defines inertia; mass is the quantitative measure of inertia.
- Newton's Second Law: $\vec{F} = m\vec{a} = \frac{\Delta \vec{p}}{\Delta t}$; force and acceleration share identical direction.
- Newton's Third Law: Action and reaction are equal and opposite, acting on two different bodies and never canceling.
- Linear momentum is $\vec{p} = m\vec{v}$; impulse is $\vec{I} = \vec{F}_{\text{avg}}\Delta t = \Delta \vec{p}$.
- In elastic collisions, momentum, total energy, and kinetic energy are conserved; in inelastic collisions, kinetic energy is lost.
- In one-dimensional elastic collisions between identical masses ($m_1 = m_2$), velocities swap completely.
- In projectile motion, horizontal velocity is constant ($v_x = v_0 \cos\theta$), while vertical motion is under constant gravity ($a_y = -g$).
- Projectile maximum height is $H = \frac{v_0^2 \sin^2\theta}{2g}$; flight time is $T = \frac{2v_0 \sin\theta}{g}$; range is $R = \frac{v_0^2 \sin(2\theta)}{g}$.
- Projectile angle relationship: $\tan\theta = \frac{4H}{R}$; maximum range occurs at 45 degrees where $R_{\max} = 4H$.
- Complementary launch angles that sum to 90 degrees ($30^\circ$ and $60^\circ$, $15^\circ$ and $75^\circ$) have identical horizontal ranges.
- Terminal velocity occurs when downward weight balances upward fluid drag force, producing zero net force and zero acceleration.