Gravity, Projectile Motion & Orbits – Exam Notes

Power

• Measure of energy delivery rate.
• Formula: $P = \frac{W}{t}$ (work per unit time).
• SI unit: watt; practical unit: horsepower.

Universal Law of Gravitation

• Newton: same force causes apples to fall & Moon’s orbit.
• $F<em>G = G \dfrac{m</em>1 m_2}{r^2}$ (masses multiply; divide by centre-to-centre distance squared).
• $G = 6.67\times10^{-11}\,\text{N·m}^2/\text{kg}^2$ (measured by Cavendish torsion-balance).

Fundamental Forces (strong→weak)

1. Strong nuclear

2. Electromagnetism

3. Weak nuclear

4. Gravity (small $G$ ⇒ weakest).

Inverse-Square Laws

• Any effect spreading radially over area $\propto r^2$ falls off as $1/r^2$.
• Gravity & light intensity are classic examples.
• Force never becomes exactly zero at finite $r$.

Apparent Weight vs Gravity

• Scale reading = support force, not direct gravity.
• Upward elevator acceleration → heavier reading; downward/free-fall → weightless (scale $=0$) though $g$ persists.
• Astronauts in ISS feel ~$0.89g$ yet are “weightless.”

Projectile Motion Basics

• Velocity vector splits into independent horizontal (no $F$) & vertical (only $mg$) parts.
• Vertical displacement: $d = \tfrac{1}{2} g t^2$.
• Drop vs horizontal launch land simultaneously (identical vertical motion).

Range & Launch Angle

• With no air drag, fixed speed gives maximum range at $45^\circ$.
• Air resistance lowers optimum to ≈$25°–34°$; even shallower for heavy, aerodynamic implements (javelin, shot-put) due to energy cost of lifting.

Orbits as Projectiles (Newton’s Cannon)

• Faster horizontal launch → projectile falls around Earth; at ~18 000 mph (≈7.8 km/s) achieves low-Earth orbit.
• Air drag converts kinetic energy to heat ⇒ satellites placed above atmosphere.

Circular vs Elliptical Orbits

• Circular: velocity always perpendicular to radial gravitational acceleration; speed constant.
• Weaker gravity at larger $r$ ⇒ lower orbital speed; leads to geosynchronous orbit at ≈$5.5R_\oplus$ (24 h period).
• Ellipse: object speeds up near periapsis, slows near apoapsis; central body sits at one focus.

Escape Speed

• Minimum surface speed to reach $r \to \infty$ with $v \to 0$:
$v<em>{esc}=\sqrt{\dfrac{2GM}{R}}$. • Values: Earth 11.2 km/s, Moon 2.4 km/s, Jupiter 60 km/s. • Greater altitude ⇒ lower required $v</em>{esc}$.
• Objects exceeding Sun’s $v_{esc}$ (e.g.
Oumuamua) are interstellar.

Key Takeaways

• Power: energy per time.
• Gravity obeys $1/r^2$, is universal yet weakest.
• Projectile paths combine independent x & y motions; ideal range max at $45^\circ$.
• Orbits are continuous projectiles; circular if speed suits local $g$, elliptical otherwise.
• Escape speed links energy to gravitational work; surpass it and an object leaves forever.