Kinematics: Freely Falling Bodies and Graphical Analysis

Freely Falling Bodies

Definition and Key Concepts

Freely falling bodies are objects that experience only the force of gravity, with all other forces (such as air resistance) considered negligible or absent. The study of their motion is fundamental in introductory physics, as it illustrates the effects of constant acceleration due to gravity.

• Gravitational Force: The only force acting on a freely falling object is gravity.

• Constant Acceleration: The acceleration of a free-falling object is constant and independent of its mass.

• Direction Convention: The upward direction is typically taken as positive in kinematic equations.

• Acceleration due to Gravity: On Earth, $g \approx -9.8\,\text{m/s}^2$ (often approximated as $-10\,\text{m/s}^2$ for calculations).

Equation for acceleration:

• $a = g$

• For velocity after time $t$: $v = v_0 + gt$

• For displacement after time $t$: $y = y_0 + v_0 t + \frac{1}{2}gt^2$

Example: An elephant and a feather, when dropped in a vacuum (no air resistance), fall with the same acceleration $g$.

Table: Velocity and Displacement of a Freely Falling Object

Time (s)	Velocity (m/s)	Displacement (m)
0	0	0
1	-10	-5
2	-20	-20
3	-30	-45

Additional info: Table entries inferred using $v = gt$ and $y = \frac{1}{2}gt^2$ with $g = -10\,\text{m/s}^2$.

Effect of Air Resistance

When air resistance is present, the net force on a falling object is reduced, and the acceleration is less than $g$. Eventually, the object reaches terminal velocity, where the net force is zero and acceleration ceases.

• Net Force: $F_{net} = W - R$ (where $W$ is weight, $R$ is air resistance)

• Terminal Velocity: When $F_{net} = 0$, $a = 0$ and the object falls at constant speed.

• Example: A skydiver’s acceleration decreases as air resistance increases, until terminal velocity is reached.

$a = \frac{F_{net}}{m} = \frac{mg - R}{m}$

Kinematic Equations and Motion Analysis

Upward and Downward Motion

When an object is thrown upward, its velocity decreases by $g$ every second until it reaches its highest point (velocity zero), then increases in the negative direction as it falls back down.

• Upward Motion: Velocity decreases by $g$ each second.

• Downward Motion: Velocity increases by $g$ each second.

• At the Highest Point: $v = 0$

Example: If a ball is thrown upward and returns after 3.0 s, its initial velocity can be found using kinematic equations.

Graphical Interpretation of Motion

Velocity and Acceleration from Graphs

Position-time and velocity-time graphs are essential tools for analyzing motion. The slope of a position-time graph gives velocity, while the slope of a velocity-time graph gives acceleration.

• Position-Time Graph: Slope = velocity ($v = \frac{d\vec{x}}{dt}$)

• Velocity-Time Graph: Slope = acceleration ($a = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}$)

• Area under Velocity-Time Graph: Represents change in position.

Example: A cyclist’s motion can be analyzed by dividing the position-time graph into segments with positive, zero, and negative velocities.

Table: Velocity and Acceleration Along Different Paths

Path	Time Interval (s)	Displacement (m)	Average Velocity (m/s)
1 (Positive velocity)	200	+400	2
2 (Zero velocity)	400	0	0
3 (Negative velocity)	400	-400	-1

Additional info: Table entries inferred from the graph and standard definitions of average velocity.

Summary of Key Equations

• Displacement: $y = y_0 + v_0 t + \frac{1}{2}gt^2$

• Velocity: $v = v_0 + gt$

• Acceleration: $a = g$ (for free fall)

• Terminal Velocity (with air resistance): $F_{net} = 0 \Rightarrow mg = R$

Applications and Examples

• Projectile Motion: The vertical and horizontal motions are independent; all objects dropped from the same height hit the ground at the same time if air resistance is negligible.

• Skydiving: Acceleration decreases as air resistance increases, leading to terminal velocity.

• Graphical Analysis: The slope and area under curves provide information about velocity and displacement.

Additional info: These notes expand on the provided slides with standard kinematic equations and graphical analysis techniques for completeness.