Free-Falling Objects and Linear Free Fall: Study Notes

Free-Falling Objects

Definition and Characteristics

Free-falling objects are those that move vertically under the influence of gravity alone, without air resistance or friction. The study of such motion is fundamental in introductory physics, as it illustrates the effects of gravity on objects near Earth's surface.

• Free fall refers to motion where gravity is the only force acting on the object.

• The acceleration due to gravity is denoted by g and is approximately constant near Earth's surface.

• Standard value: $g = 9.80\ \mathrm{m/s^2}$

• Direction: Acceleration due to gravity always points toward the center of the Earth (downward).

• Air resistance is ignored in ideal free-fall problems.

• Altitude effect: The value of $g$ decreases slightly with altitude (e.g., it is less at 10,000 m above sea level than at the surface).

Additional info: The value of $g$ can vary slightly depending on latitude and local geological formations, but $9.80\ \mathrm{m/s^2}$ is used for most calculations.

Acceleration Due to Gravity

Sign Convention and Application

When analyzing free-fall motion, it is important to define the coordinate system. Typically, the upward direction is taken as positive, making the acceleration due to gravity negative.

• Acceleration in free fall: $a_y = -g = -9.80\ \mathrm{m/s^2}$ (if upward is positive)

• For an object at rest: $v_y = 0$, but $a_y = -g$ (it will begin to move downward).

• For an object moving upward: $a_y = -g$ (it slows down due to gravity).

• For an object moving downward: $a_y = -g$ (it speeds up due to gravity).

Key Point: The acceleration due to gravity is always downward and constant for free-falling objects near Earth's surface.

Equations for Linear Free Fall

Kinematic Equations and Their Use

The motion of free-falling objects can be described using the kinematic equations for constant acceleration. These equations relate displacement, velocity, acceleration, and time.

• Coordinate system: If the y-axis points up, then $a_y = -g$.

• Common variables: $y$ (position), $y_0$ (initial position), $v_y$ (velocity), $v_{0y}$ (initial velocity), $t$ (time).

Equation	Description	Variables Included
$v_y = v_{0y} - gt$	Velocity as a function of time (linear)	$v_y$, $v_{0y}$, $t$
$y = y_0 + v_{0y}t - \frac{1}{2}gt^2$	Position as a function of time (parabolic)	$y$, $y_0$, $v_{0y}$, $t$
$v_y^2 = v_{0y}^2 - 2g(y - y_0)$	Velocity as a function of displacement	$v_y$, $v_{0y}$, $y$, $y_0$

How to use: Choose the equation that includes the known and unknown variables for the problem at hand.

Example Application

• Example: A ball is thrown upward with an initial velocity $v_{0y}$. At its highest point, $v_y = 0$, and you can use $v_y = v_{0y} - gt$ to solve for the time to reach the maximum height.

• Example: If a rock is dropped from rest ($v_{0y} = 0$) from a height $y_0$, its position after time $t$ is $y = y_0 - \frac{1}{2}gt^2$.

Additional info: These equations are valid only when acceleration is constant (i.e., air resistance is negligible).