Kinematics: The Geometry of Motion

Understanding how things move without concerning ourselves with why they move.

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause the motion. It is purely descriptive.

The Big Four Equations

For constant acceleration ($a$), linear motion can be fully described by four equations derived from calculus. These relate displacement ($\Delta x$), initial velocity ($v_0$), final velocity ($v$), acceleration ($a$), and time ($t$).

Equation	Missing Variable
$$ v = v_0 + at $$	Displacement ($\Delta x$)
$$ \Delta x = \frac{v_0 + v}{2}t $$	Acceleration ($a$)
$$ \Delta x = v_0 t + \frac{1}{2}at^2 $$	Final Velocity ($v$)
$$ v^2 = v_0^2 + 2a\Delta x $$	Time ($t$)

Worked Example: Dropped Object

Problem: A rock is dropped from a 50-meter cliff. How long does it take to hit the ground, and what is its final velocity? (Assume no air resistance and $g = 9.81 \, \text{m/s}^2$).

Solution:

1. Identify knowns: $v_0 = 0$, $\Delta y = -50 \, \text{m}$, $a = -9.81 \, \text{m/s}^2$.
2. To find time ($t$), use $\Delta y = v_0 t + \frac{1}{2}at^2$.
3. $-50 = 0 + \frac{1}{2}(-9.81)t^2$
4. $-50 = -4.905 t^2 \implies t^2 = 10.19 \implies t \approx 3.19 \, \text{s}$.
5. To find final velocity ($v$), use $v = v_0 + at$.
6. $v = 0 + (-9.81)(3.19) \approx -31.3 \, \text{m/s}$.

Common Mistakes

• Sign Errors: Forgetting that displacement, velocity, and acceleration are vectors. If up is positive, gravity must be negative ($-9.8 \text{m/s}^2$).
• Using constant acceleration equations when acceleration varies: These equations only work when $a$ is constant. For variable acceleration, calculus must be used ($v = \int a \, dt$).