Kinematics Explained

Kinematics is the branch of physics that describes the motion of objects without considering the forces causing that motion. It focuses on variables like displacement, velocity, acceleration, and time, providing a mathematical framework to analyze how objects move in one or two dimensions. Kinematics is foundational for understanding more complex topics like dynamics and is widely applied in fields such as engineering, animation, and robotics. This guide covers all basic kinematic equations, including linear and projectile motion, relative motion, and provides detailed examples and applications to illustrate these concepts.

Motion Equations (with Relative Motion, Projectile Motion)

Kinematics uses equations to describe motion under constant acceleration. These equations relate displacement, velocity, acceleration, and time. For two-dimensional motion, such as projectiles, we break motion into horizontal and vertical components.

Basic Formulas (Uniformly Accelerated Motion):

  • 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 Motion (Horizontal): \( x = u_x t \), \( u_x = u \cos\theta \)
  • Projectile Motion (Vertical): \( y = u_y t - \frac{1}{2} g t^2 \), \( u_y = u \sin\theta \)
  • Time of Flight: \( T = \frac{2u \sin\theta}{g} \)
  • Range: \( R = \frac{u^2 \sin(2\theta)}{g} \)

Where:

  • \( s \): Displacement (m)
  • \( u \): Initial velocity (m/s)
  • \( v \): Final velocity (m/s)
  • \( a \): Acceleration (m/s²)
  • \( t \): Time (s)
  • \( g \): Gravitational acceleration (\( 9.8 \, \text{m/s}^2 \))
  • \( \theta \): Launch angle (degrees)

Examples

Example 1: Free Fall

A ball dropped from rest (\( u = 0 \)) with \( a = 9.8 \, \text{m/s}^2 \) for 2 seconds:

\[ s = ut + \frac{1}{2} a t^2 \] \[ = 0 \cdot 2 + \frac{1}{2} \cdot 9.8 \cdot (2)^2 \] \[ = 0 + \frac{1}{2} \cdot 9.8 \cdot 4 \] \[ = 4.9 \cdot 4 \] \[ = 19.6 \, \text{m} \]

Example 2: Final Velocity

A car accelerates from \( u = 5 \, \text{m/s} \) at \( a = 2 \, \text{m/s}^2 \) for 3 seconds:

\[ v = u + at \] \[ = 5 + 2 \cdot 3 \] \[ = 5 + 6 \] \[ = 11 \, \text{m/s} \]

Example 3: Velocity-Displacement

A bike starts from rest (\( u = 0 \)), accelerates at \( a = 3 \, \text{m/s}^2 \), travels \( s = 12 \, \text{m} \). Find final velocity:

\[ v^2 = u^2 + 2as \] \[ = 0^2 + 2 \cdot 3 \cdot 12 \] \[ = 0 + 72 \] \[ v^2 = 72 \] \[ v = \sqrt{72} \] \[ \approx 8.49 \, \text{m/s} \]

Example 4: Average Velocity

A runner with \( u = 4 \, \text{m/s} \), \( v = 8 \, \text{m/s} \). Find average velocity:

\[ v_{\text{avg}} = \frac{u + v}{2} \] \[ = \frac{4 + 8}{2} \] \[ = \frac{12}{2} \] \[ = 6 \, \text{m/s} \]

Example 5: Relative Velocity

Car A moves at \( v_A = 20 \, \text{m/s} \), Car B at \( v_B = 15 \, \text{m/s} \) in the same direction. Find relative velocity of A with respect to B:

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

Example 6: Projectile Motion (Range)

A ball launched at \( u = 30 \, \text{m/s} \), \( \theta = 45^\circ \), \( g = 9.8 \, \text{m/s}^2 \). Find range:

\[ R = \frac{u^2 \sin(2\theta)}{g} \] \[ \sin(2 \cdot 45^\circ) = \sin(90^\circ) = 1 \] \[ R = \frac{(30)^2 \cdot 1}{9.8} \] \[ = \frac{900}{9.8} \] \[ \approx 91.84 \, \text{m} \]

Example 7: Projectile Motion (Time of Flight)

Same ball (\( u = 30 \, \text{m/s} \), \( \theta = 45^\circ \), \( g = 9.8 \, \text{m/s}^2 \)). Find time of flight:

\[ T = \frac{2u \sin\theta}{g} \] \[ \sin(45^\circ) \approx 0.707 \] \[ T = \frac{2 \cdot 30 \cdot 0.707}{9.8} \] \[ = \frac{42.42}{9.8} \] \[ \approx 4.33 \, \text{s} \]

Applications

Kinematics is crucial in various fields. Below are examples with calculations:

Example 1: Animation (Character Jump)

A character jumps with \( u = 5 \, \text{m/s} \), \( a = -9.8 \, \text{m/s}^2 \). Find height after 0.3 s:

\[ s = ut + \frac{1}{2} a t^2 \] \[ = 5 \cdot 0.3 + \frac{1}{2} \cdot (-9.8) \cdot (0.3)^2 \] \[ = 1.5 - 4.9 \cdot 0.09 \] \[ = 1.5 - 0.441 \] \[ \approx 1.06 \, \text{m} \]

Example 2: Robotics (Arm Motion)

A robotic arm moves with \( u = 2 \, \text{m/s} \), \( a = 4 \, \text{m/s}^2 \), for 1 s. Find displacement:

\[ s = ut + \frac{1}{2} a t^2 \] \[ = 2 \cdot 1 + \frac{1}{2} \cdot 4 \cdot (1)^2 \] \[ = 2 + 2 \] \[ = 4 \, \text{m} \]

Example 3: Vehicle Design (Braking Distance)

A car at \( u = 20 \, \text{m/s} \) decelerates at \( a = -5 \, \text{m/s}^2 \). Find stopping distance:

\[ v^2 = u^2 + 2as \] \[ 0 = (20)^2 + 2 \cdot (-5) \cdot s \] \[ 0 = 400 - 10s \] \[ 10s = 400 \] \[ s = 40 \, \text{m} \]

Example 4: Sports (Basketball Shot)

A basketball shot at \( u = 10 \, \text{m/s} \), \( \theta = 60^\circ \). Find range:

\[ R = \frac{u^2 \sin(2\theta)}{g} \] \[ \sin(2 \cdot 60^\circ) = \sin(120^\circ) \approx 0.866 \] \[ R = \frac{(10)^2 \cdot 0.866}{9.8} \] \[ = \frac{86.6}{9.8} \] \[ \approx 8.84 \, \text{m} \]

Example 5: Aviation (Takeoff)

A plane accelerates from rest at \( a = 3 \, \text{m/s}^2 \) for 10 s. Find velocity:

\[ v = u + at \] \[ = 0 + 3 \cdot 10 \] \[ = 30 \, \text{m/s} \]

Example 6: Train Motion (Relative Velocity)

Train A moves at 30 m/s, Train B at 20 m/s in opposite directions. Find relative velocity:

\[ v_{AB} = v_A - v_B \] \[ = 30 - (-20) \] \[ = 30 + 20 \] \[ = 50 \, \text{m/s} \]