Gravitational Acceleration and Weight

Near Earth's surface, all objects experience a downward gravitational acceleration of:<br>$$g = 9.81\ \text{m/s}^2$$

This value is essentially constant close to the surface (it decreases slightly at higher altitudes). Weight is the gravitational force exerted on an object:<br>$$W = mg$$<br>where $W$ is weight (N), $m$ is mass (kg), and $g = 9.81\ \text{m/s}^2$.

Example: A person of mass 65 kg has a weight of $W = 65 \times 9.81 = 637.65\ \text{N}$ on Earth's surface.

Important distinction: Mass is the amount of matter in an object (kg) — it is the same everywhere. Weight is a force that depends on gravitational field strength (N) — it varies by location. An astronaut on the Moon weighs about one-sixth of their Earth weight, but their mass is unchanged.

Free Fall

An object in free fall is subject only to gravity — air resistance is neglected. In free fall, every object accelerates downward at $g = 9.81\ \text{m/s}^2$, regardless of its mass. This was Galileo's great insight, confirmed centuries later in vacuum experiments.

Key free-fall equations (object starting from rest, taking downward as positive):<br>$$v = gt$$<br>$$h = \frac{1}{2}gt^2$$<br>$$v^2 = 2gh$$

Example: An object is dropped from rest. After 3 s:

• Speed: $v = 9.81 \times 3 = 29.43\ \text{m/s}$
• Distance fallen: $h = \frac{1}{2} \times 9.81 \times 3^2 = 44.1\ \text{m}$

Projectile Motion

A projectile is an object launched into the air subject only to gravity (air resistance neglected). The key principle is the independence of horizontal and vertical motion:

• Horizontal: No net horizontal force acts on the projectile, so horizontal velocity $v_x$ remains constant. Horizontal distance: $x = v_x t$.
• Vertical: Gravity causes a constant downward acceleration $g$, described by the same free-fall equations.

Example: A ball is thrown horizontally from a cliff at $10\ \text{m/s}$ and takes 2 s to reach the ground.

• Horizontal distance: $x = 10 \times 2 = 20\ \text{m}$
• Vertical drop: $h = \frac{1}{2} \times 9.81 \times 2^2 = 19.6\ \text{m}$

A key consequence: a ball thrown horizontally and a ball simply dropped from the same height will hit the ground at exactly the same time, because their vertical motions are identical.