Time and Motion

A body is in motion if its position changes relative to a reference frame over time; otherwise, it is at rest. All rest and motion are relative; there is no absolute rest in the cosmos.

• Rectilinear: Straight line path (falling apple).
• Circular: Path along circular arc about external point (clock hands).
• Rotational: Spinning about internal axis (spinning top).
• Periodic/Oscillatory: To-and-fro motion repeating at fixed intervals (pendulum).
• Compound/Mixed: Combination of translation and rotation (rolling bicycle wheel).

A point on the blade of a ceiling fan executes circular motion, while the fan body as an entity executes rotational motion.

Rotational motion has the axis passing through the body; circular motion has the axis outside the body.

Galileo’s ship thought experiment proved that inside a smoothly sailing vessel, no mechanical test can detect uniform motion without looking outside.

Distance: Total actual path length traversed (scalar quantity). Displacement: Shortest directed straight-line segment from initial to final position (vector quantity).

Distance is always positive ($d > 0$) for moving bodies. Displacement can be positive, negative, or zero. Distance $\geq$ |Displacement|; equality holds only for unidirectional rectilinear motion.

In one complete circle of radius $r$: Distance $= 2\pi r$; Displacement $= 0$ because starting and ending points coincide.

Zero displacement does NOT imply zero distance traveled! Returning home from school yields zero displacement but non-zero distance.

Both distance and displacement share the SI unit meter (m) and CGS unit centimeter (cm).

Speed: Distance traveled per unit time ($v = s/t$, scalar). Velocity: Displacement per unit time in a specified direction ($\vec{v} = \vec{s}/t$, vector).

Average speed $= \frac{\text{Total Distance}}{\text{Total Time}}$. It accounts for variable speeds over a full journey.

$1\text{ km/h} = \frac{1000\text{ m}}{3600\text{ s}} = \frac{5}{18}\text{ m/s}$. Example: $54\text{ km/h} = 54 \times \frac{5}{18} = 15\text{ m/s}$.

Traveling out at $40\text{ km/h}$ and returning at $60\text{ km/h}$ gives an average speed of $\frac{2 \times 40 \times 60}{40 + 60} = 48\text{ km/h}$, NOT $50\text{ km/h}$, because more time is spent at lower speed.

Speedometers display instantaneous speed; Odometers record total cumulative distance traveled.

Acceleration is the rate of increase of velocity with time. Retardation (Deceleration) is the rate of decrease of velocity (negative acceleration).

$$a = \frac{v - u}{t} \iff v = u + at$$ For retardation: $v = u - at$.

SI unit: $\text{m/s}^2$. Acceleration $= \frac{\text{Change in velocity (m/s)}}{\text{Time (s)}}$. The first "second" indicates velocity units; the second indicates rate of change.

Bodies falling freely near Earth accelerate downward at $g \approx 9.8\text{ m/s}^2$ regardless of mass in vacuum.

Can a body moving at constant speed have acceleration? Yes! A body in uniform circular motion constantly changes direction, generating centripetal acceleration.

Every body continues in its state of rest or uniform rectilinear motion unless acted upon by an external net force. Defines inertia (inertia of rest vs inertia of motion).

Rate of change of momentum is proportional to applied force: $F = m \cdot a$. SI unit: Newton (N), CGS: Dyne ($1\text{ N} = 10^5\text{ Dynes}$).

To every action there is an equal and opposite reaction ($F_1 = -F_2$). Seen in walking, swimming, rocket propulsion, and rifle recoil.

Friction opposes relative motion between contacting surfaces. Without friction, walking, driving, and braking would be impossible.

Fielders draw hands backward when catching balls to prolong stopping time ($t$), reducing impact force ($F = \Delta p / t$) on hands.