Kinematics

Kinematics, the study of motion without forces, analyzes displacement, velocity, acceleration, and time. Essential in physics, it underpins applications in engineering, robotics, and animation. This MathMultiverse guide explores kinematic equations, including linear and projectile motion, with examples, visualizations, and real-world applications.

Why kinematics? It provides precise tools to model and predict motion, foundational for dynamics and practical design.

Motion Equations

Kinematics describes motion with equations for constant acceleration, handling linear and two-dimensional cases like projectiles.

Key Equations:

  • Displacement: \( s = ut + \frac{1}{2} a t^2 \)
  • Final Velocity: \( v = u + at \)
  • Velocity-Displacement: \( v^2 = u^2 + 2as \)
  • Average Velocity: \( v_{\text{avg}} = \frac{u + v}{2} \)
  • Relative Velocity: \( v_{AB} = v_A - v_B \)
  • Projectile (Horizontal): \( x = u \cos\theta \, t \)
  • Projectile (Vertical): \( y = u \sin\theta \, t - \frac{1}{2} g t^2 \)
  • Time of Flight: \( T = \frac{2u \sin\theta}{g} \)
  • Range: \( R = \frac{u^2 \sin(2\theta)}{g} \)

Variables: \( s \) (displacement, m), \( u \) (initial velocity, m/s), \( v \) (final velocity, m/s), \( a \) (acceleration, m/s²), \( t \) (time, s), \( g = 9.8 \, \text{m/s}^2 \), \( \theta \) (angle, degrees).

Projectile Motion Visualization

Trajectory for \( u = 30 \, \text{m/s} \), \( \theta = 45^\circ \).

Examples

1. Free Fall

Ball dropped (\( u = 0 \), \( a = 9.8 \, \text{m/s}^2 \), \( t = 2 \, \text{s} \)):

\[ s = 0 \cdot 2 + \frac{1}{2} \cdot 9.8 \cdot 2^2 = 19.6 \, \text{m} \]

2. Final Velocity

Car (\( u = 5 \, \text{m/s} \), \( a = 2 \, \text{m/s}^2 \), \( t = 3 \, \text{s} \)):

\[ v = 5 + 2 \cdot 3 = 11 \, \text{m/s} \]

3. Velocity-Displacement

Bike (\( u = 0 \), \( a = 3 \, \text{m/s}^2 \), \( s = 12 \, \text{m} \)):

\[ v = \sqrt{2 \cdot 3 \cdot 12} \approx 8.49 \, \text{m/s} \]

4. Average Velocity

Runner (\( u = 4 \, \text{m/s} \), \( v = 8 \, \text{m/s} \)):

\[ v_{\text{avg}} = \frac{4 + 8}{2} = 6 \, \text{m/s} \]

5. Relative Velocity

Car A (\( v_A = 20 \, \text{m/s} \)), Car B (\( v_B = 15 \, \text{m/s} \)):

\[ v_{AB} = 20 - 15 = 5 \, \text{m/s} \]

6. Projectile Range

Ball (\( u = 30 \, \text{m/s} \), \( \theta = 45^\circ \)):

\[ R = \frac{30^2 \cdot 1}{9.8} \approx 91.84 \, \text{m} \]

7. Time of Flight

Ball (\( u = 30 \, \text{m/s} \), \( \theta = 45^\circ \)):

\[ T = \frac{2 \cdot 30 \cdot 0.707}{9.8} \approx 4.33 \, \text{s} \]

Applications

Kinematics drives real-world solutions:

  • Animation: Character jump (\( u = 5 \, \text{m/s} \), \( t = 0.3 \, \text{s} \)):
    \[ s = 5 \cdot 0.3 - 4.9 \cdot 0.09 \approx 1.06 \, \text{m} \]
  • Robotics: Arm motion (\( u = 2 \, \text{m/s} \), \( a = 4 \, \text{m/s}^2 \), \( t = 1 \, \text{s} \)):
    \[ s = 2 \cdot 1 + 2 \cdot 1^2 = 4 \, \text{m} \]
  • Vehicle Design: Braking (\( u = 20 \, \text{m/s} \), \( a = -5 \, \text{m/s}^2 \)):
    \[ s = \frac{20^2}{2 \cdot 5} = 40 \, \text{m} \]
  • Sports: Basketball shot (\( u = 10 \, \text{m/s} \), \( \theta = 60^\circ \)):
    \[ R = \frac{10^2 \cdot 0.866}{9.8} \approx 8.84 \, \text{m} \]
  • Aviation: Takeoff (\( u = 0 \), \( a = 3 \, \text{m/s}^2 \), \( t = 10 \, \text{s} \)):
    \[ v = 3 \cdot 10 = 30 \, \text{m/s} \]
  • Train Motion: Relative velocity (\( v_A = 30 \, \text{m/s} \), \( v_B = -20 \, \text{m/s} \)):
    \[ v_{AB} = 30 + 20 = 50 \, \text{m/s} \]