Chapter 3: Linear Motion — Comprehensive Notes

Relative Motion and Foundations

Everything in motion is described relative to something else (e.g., Earth, Sun, track, or a moving bus).
Examples:
- Sitting on a chair: your speed is 0 relative to Earth, but nonzero relative to the Sun due to Earth’s motion.
- A person on a bus and another plane flying in opposite direction: observed speeds change depending on reference frame; relative motion can double when comparing two objects moving toward each other.
Motion Is Relative: speeds given in the text are typically relative to Earth unless stated otherwise.
Practical illustration: If you walk down the aisle of a moving bus, your speed relative to the bus floor differs from your speed relative to the road.

Speed, Velocity, and Units

Speed is how fast an object moves, defined as distance per unit time.
Velocity is speed with direction; it is a vector quantity.
Speed is a scalar quantity; velocity is a vector quantity.
Speed definitions and units:
- Speed = distance ÷ time; unit example: ext{Speed} = \frac{d}{t}
- Instantaneous speed vs average speed:
- Instantaneous speed is the speed at an exact moment.
- Average speed is the total distance divided by the total time: \bar{v} = \frac{\text{total distance}}{\text{time interval}}
Common speed units and conversions (from the transcript):
- 12 mph = 20 km/h = 6 m/s
- 25 mph = 40 km/h = 11 m/s
- 37 mph = 60 km/h = 17 m/s
- 50 mph = 80 km/h = 22 m/s
- 62 mph = 100 km/h = 28 m/s
- 75 mph = 120 km/h = 33 m/s
- 100 mph = 160 km/h = 44 m/s
Relative motion example: If you sit on a chair, your speed is 0 relative to Earth but not to the Sun; relative motion is the general rule.

Speed vs Velocity in Motion Paths

Constant speed vs constant velocity:
- Constant speed means speed remains the same, but direction may change (e.g., car on a circular track).
- Constant velocity means constant speed and straight-line motion (no change in velocity).
Velocity changes if either speed or direction changes (or both).
A curved-path motion can have constant speed but non-constant velocity due to changing direction.
Check Your Understanding (concept focuses):
- If a car moves with a constant speed but along a curved path, its velocity is not constant.
- If two cars move with the same speed but in opposite directions, their velocities have the same magnitude but opposite directions.

Acceleration: Change of Velocity

Acceleration is the rate of change of velocity: a = \frac{\Delta v}{\Delta t}
Change in velocity can be from speed changes, direction changes, or both.
Examples:
- Slamming on the accelerator increases velocity; the passengers feel acceleration.
- Braking produces a deceleration (negative acceleration).
Along a straight line, one often writes acceleration as: a = \frac{vf - vi}{t}
Three key observations:
- Acceleration can be positive (speeding up), negative (slowing down), or zero.
- Velocity and acceleration are related but distinct: velocity is a rate of change of position; acceleration is a rate of change of velocity.
- In circular motion with constant speed, acceleration is present due to changes in direction (centripetal acceleration).
Galileo’s contribution to acceleration: introduced the concept of constant acceleration on inclined planes; the gain in speed per second is constant for a given incline.

Galileo, Inclined Planes, and Free Fall

Galileo’s motion studies laid the foundation for Newtonian mechanics:
- He investigated motion on inclined planes and discovered that objects gain the same amount of speed per second on a given incline.
- The speed gained per second is acceleration; for steeper inclines, acceleration is greater.
- When an incline is vertical, the acceleration equals the acceleration due to gravity for freely falling objects (neglecting air resistance).
Free fall: Motion under gravity with no air resistance.
- Free-fall acceleration on Earth is denoted by g.
- Average values used in teaching: g \approx 9.8\ \text{m/s}^2 (often rounded to 10 for simplicity in introductory work).
- The velocity gained in free fall from rest after time t is: v(t) = g t
- The distance fallen from rest after time t is: d(t) = \tfrac{1}{2} g t^2
- For Earth, g is roughly constant in many problems; Moon and other planets have different g values.
In air, heavier objects fall essentially with the same acceleration as others when air resistance is small; air resistance can cause different accelerations for light objects (feather, coin) compared to heavier objects (stone, baseball).
The velocity-time relation for a freely falling object from rest w/ no air resistance can be summarized as:
- Instantaneous velocity: v = g t
- Distance fallen: d = \tfrac{1}{2} g t^2
Velocity vs. time relationships during vertical motion:
- If an object is thrown upward, its velocity decreases by about 10 m/s each second (on Earth) until it reaches zero at the top, then increases in the opposite direction on the way down.
- The acceleration due to gravity remains constant (≈ 10 m/s², or 9.8 m/s² in precise calculations) downward throughout the motion.

Distance and Velocity in Uniform Acceleration

Distance fallen in uniform acceleration when starting from rest:
- d = \tfrac{1}{2} a t^2
- For gravity, with initial velocity zero: d = \tfrac{1}{2} g t^2
For a general motion with initial velocity v0:
- Velocity: v = v_0 + a t
- Distance: d = v_0 t + \tfrac{1}{2} a t^2
Relationship between velocity acquired and time on an inclined plane:
- On a given incline, velocity gained per unit time is proportional to the slope; steeper incline → greater acceleration.
- For vertical fall, acceleration equals g and is constant in the absence of air resistance.
The idea that velocity and acceleration can be plotted as functions of time leads to kinematic equations used across problems.

Hang Time and Jumping Vertically

Hang time: time a jumper remains airborne.
Vertical jump physics:
- The upward launch speed determines how high the jumper rises and the total hang time.
- An example calculation: if the maximum vertical height is h, then the time to reach the top is t{up} = \sqrt{\frac{2h}{g}}, and the total hang time is T = 2 t{up} = 2\sqrt{\frac{2h}{g}}.
- With a standing vertical jump height of about 1.25 m and g ≈ 9.8 m/s², the total hang time is approximately 1 s.
Large jumps (e.g., basketball players) demonstrate that hang time is limited by vertical speed at takeoff, not by leg muscle alone; air resistance is negligible in these idealized calculations.
The horizontal speed during a jump does not affect hang time (hang time depends on vertical velocity and gravity).
Key takeaway: hang time and maximum height depend on the initial vertical speed and g; velocity changes by a constant of -g during the ascent and +g during descent (signs indicate direction).

Practical Equations and Tables (from the text)

Speed and velocity conversions:
- A quick reference table includes conversions among mph, km/h, and m/s (as shown above).
Key equations (summary):
- Speed: v = \frac{d}{t}
- Average speed: \bar{v} = \frac{D}{\Delta t}
- Velocity: \vec{v} = \frac{\Delta \vec{x}}{\Delta t}
- Acceleration: a = \frac{\Delta v}{\Delta t}
- Straight-line acceleration when starting from rest: d = \tfrac{1}{2} a t^2
- Velocity from rest under constant acceleration: v = a t (for straight-line, starting from rest)
- Free fall velocity: v = g t
- Free fall distance: d = \tfrac{1}{2} g t^2
- Distance fallen under constant acceleration (general): d = v_0 t + \tfrac{1}{2} a t^2
- Velocity acquired on incline: v = a t (where a depends on incline angle)
- Hang time for a jump with height h: T = 2\sqrt{\frac{2h}{g}}
Notation and constants:
- g is acceleration due to gravity, approximately g \approx 9.8\ \text{m/s}^2 (often rounded to 10 in introductory contexts).
- Time is in seconds (s), speed in m/s, distance in meters (m).
Important distinctions:
- Velocity is the rate of change of position with direction; acceleration is the rate of change of velocity.
- Zero acceleration does not necessarily mean no motion (could be constant velocity); zero velocity means you are at rest.
- An object can have constant speed but changing velocity if its direction changes (e.g., turning in a circle).

Worked Examples and Checkpoints (Conceptual)

Check Your Answers style prompts encourage predicting the outcome before reading solutions to promote understanding. Examples include:
- If a car moves from rest to a high speed, its acceleration is the rate of speed increase, not just the final speed.
- When comparing two objects with the same instantaneous speed but different directions, their velocities differ in direction but may have the same magnitude.
- A car moving at a constant velocity on a curved road has changing velocity due to the changing direction; its acceleration is not zero.
Applications of the core formulas to practice problems (as given in the transcript):
- Average speed problems (distance and time provided).
- Instantaneous vs average speed inferred from a speedometer reading.
- Free-fall problems with given times to compute velocity and distance.
- Distance traveled during uniform acceleration using the formula d = \tfrac{1}{2} a t^2 or d = v_0 t + \tfrac{1}{2} a t^2 depending on the initial velocity.
- Hang-time and vertical jump problems using T = 2\sqrt{2h/g} or equivalent derivations.

Summary of Key Concepts (Condensed)

Speed vs Velocity:
- Speed: how fast, magnitude only (scalar).
- Velocity: how fast and in what direction (vector).
Acceleration:
- Rate of change of velocity; can be due to speed changes, direction changes, or both.
- Positive acceleration speeds up in the same direction; negative acceleration (deceleration) slows down.
Equations (core toolkit):
- v = \frac{d}{t}
- \bar{v} = \frac{D}{\Delta t}
- \vec{v} = \frac{\Delta \vec{x}}{\Delta t}
- a = \frac{\Delta v}{\Delta t}
- d = v_0 t + \tfrac{1}{2} a t^2 (general)
- d = \tfrac{1}{2} a t^2 (from rest)
- v = v_0 + a t
- For gravity: v = g t, \quad d = \tfrac{1}{2} g t^2, \quad g \approx 9.8\ \text{m/s}^2
- Hang time for a jump: T = 2\sqrt{\frac{2h}{g}}
Conceptual links to real-world relevance:
- All motion is described relative to a reference frame; no absolute motion.
- Real-world motion often involves curved paths where velocity changes even if speed is constant.
- Understanding these concepts helps in sports, driving, aviation, and physics analysis of everyday phenomena.

Quick Practice Prompts (from the transcript prompts)

A) If a car moves with an average speed of $50$ km/h for $1$ hour, what distance does it travel? If the average speed is the same for $4$ hours, what distance?
B) Does a constant speedometer reading imply constant speed or constant velocity?
C) On a horizontal track, two cars moving with the same speed but opposite directions have what relation between their velocities?
D) If air resistance is neglected, what is the acceleration of a freely falling object on Earth? How does it change with time?
E) How is hang time related to jump height and gravity?
F) Given a height $h$, derive the time to reach the peak and the total hang time for vertical motion under gravity.

Appendix: Notation and Key Tables (as referenced)

Table 3.1: Approximate speeds in different units (illustrative values listed in the text).
Table 3.2: Instantaneous speed for freely falling object at 1-second intervals (speeds increase by about 10 m/s every second when g ≈ 10 m/s²).
Table 3.3: Distance fallen in free fall for successive seconds (e.g., 1 s → 5 m, 2 s → 20 m, 3 s → 45 m, 4 s → 80 m, 5 s → 125 m, etc.).
Note: In practice, more precise calculations use $g = 9.8\ \text{m/s}^2$, but $g = 10\ \text{m/s}^2$ is used for clarity in many demonstrations.

Note on Real-World Relevance and Ethics

The historical discussion of Galileo and his conflict with the Church underscores the ethical dimensions of science: the pursuit of knowledge can challenge established doctrine, and evidence must guide belief.
The concept of relative motion is foundational for technologies such as GPS, aerospace navigation, and any system that uses reference frames.
Understanding acceleration and free fall informs safety features (airbags, braking systems, skydiver safety) and helps interpret sports performance and design.