Notes on Free Fall and Projectile Motion (Transcript)

Question (Page 1): If two objects with different masses are thrown from the same height (3rd floor), which will reach the ground first?
Answer implied by Galileo: In the absence of air resistance, they reach ground at the same time.
Context: Early ideas about motion include Aristotle’s views, Galileo’s experiments, and Newtonian gravity.

Terrestrial motion depends on the material contained in the object.
A stone thrown upward returns to Earth because it contains so much, namely “earth.”
He also asserted that heavier objects fall faster than lighter ones.

Galileo questioned Aristotle’s ideas about falling bodies.
Experiment: rolled steel balls of different weights down an inclined plane.
Measured the time it took for balls to reach markings on the plane and at the bottom.
Key observation: the bottom reached in almost the square of the elapsed time, implying a constant acceleration and that heavier or lighter balls fall at the same rate when air resistance is negligible.
Conclusion: Without air resistance, all objects would fall to the ground at the same rate.

Objects fall toward the ground due to gravity (the force pulling toward Earth’s center).
Free fall: motion of a body influenced only by gravity.
Acceleration due to gravity: a = -g \text{ with } g \approx 9.8 \,\mathrm{m/s^2}
In vertical motion, the displacement d is often written as dy, and acceleration as g in the equations.

Example 1 (Page 12): Ball falls from rest for 5.0 s.
- Given: v_0 = 0,\ a = -g = -9.8 \,\mathrm{m/s^2},\ t = 5.0\,\mathrm{s}
- Velocity: v = v_0 + a t = 0 + (-9.8)(5.0) = -49 \,\mathrm{m/s}
Example 2 (Page 13): Rock dropped from a cliff, no air resistance, after 6.0 s.
- Velocity: v = 0 + (-9.8)(6.0) = -58.8 \,\mathrm{m/s}
Example 3 (Page 14): Object dropped from a bridge reaches v = -29.4 \,\mathrm{m/s}; find time.
- Solve: -29.4 = 0 + (-9.8) t \Rightarrow t = \frac{29.4}{9.8} = 3.0 \mathrm{s}

Projectile: any object thrown or moved with an initial velocity and acted upon by gravity.
Trajectory: the path that the projectile follows.
In projectile motion, gravity causes a downward acceleration, leading to a curved (parabolic) trajectory.
Vertical and horizontal components of motion can be treated separately due to independence of motions in perpendicular directions.

Initial speed and launch angle: U is the launch speed, and \theta is the launch angle.
Horizontal component: u_x = U \,\cos \theta
- Note in LaTeX: u_x = U \cos \theta but in the transcript the slash is escaped as U \cos \theta in plain text to render correctly in Markdown.
Vertical component: u_y = U \,\sin \theta
Vertical acceleration: a_y = -g
Range, horizontal extent, depends on time of flight and horizontal velocity.
If launch and landing heights are the same, horizontal range is given by: R = \dfrac{U^2 \sin(2\theta)}{g}
Range is maximized at \theta = 45^\circ (for level ground without air resistance).

Effect of angle on range:
- As the angle increases from 0° to 45°, the range increases.
- Beyond 45°, the range decreases.
Effect of angle on maximum height:
- Height increases with angle up to 90° (vertical launch).
- At 90°, the height is maximized but the range is zero.

Historical progression: Aristotle (heavier objects fall faster) → Galileo (experimentally showed equal rates without air resistance) → Newton (gravity governs motion; constant downward acceleration).
Real-world relevance: Air resistance affects real objects; in vacuum, different masses fall at the same rate.
Foundational principles: Uniform acceleration, vector components, independence of motions along perpendicular directions, and simple projectile motion formulas.
Ethical/philosophical notes: The shift from qualitative to quantitative science; using experiments (inclined planes) to test theories; acknowledging and controlling for air resistance in experiments.

Constant acceleration relation:
v = v_0 + a t
Acceleration due to gravity (downward):
a = -g = -9.8 \ \mathrm{m/s^2}
Vertical displacement under gravity (common form):
y = y0 + v{0y} t - \tfrac{1}{2} g t^2
Horizontal displacement (constant horizontal velocity):
x = x0 + v{0x} t
Projectile launch components:
- u_x = U \cos \theta
- u_y = U \sin \theta
Projectile vertical acceleration:
- a_y = -g
Projectile range (level ground, no air resistance):
R = \dfrac{U^2 \sin(2\theta)}{g}
Time of flight (approximate for level ground):
T = \dfrac{2 U \sin \theta}{g}
Maximum height (vertical launch):
H = \dfrac{U^2 \sin^2 \theta}{2 g}
Sign conventions (free-fall conventions):
- Distances above origin: positive; below origin: negative
- Upward velocity: positive; downward velocity: negative
- Acceleration due to gravity: negative

In vacuum, all bodies accelerate at the same rate under gravity, independent of mass.
Real-world experiments must account for air resistance, which can cause heavier objects to appear to fall faster in air.
The horizontal and vertical motions in projectile motion evolve independently but combine to produce a parabolic trajectory.
Angles near 0° produce long ranges but low heights; angles near 90° produce high heights but short ranges.