Kinematics

Kinematics

Kinematics describes the motion of objects without considering the forces that cause the motion. It involves concepts such as displacement, velocity, and acceleration.

Motion in 1D

Displacement, Velocity, and Acceleration

Displacement ($\Delta x$): The change in position of an object. It is a vector quantity. $\Delta x = x_f - x_i$

Velocity ($\mathbf{v}$): The rate of change of displacement.

• Average Velocity: $\bar{v} = \frac{\Delta x}{\Delta t}$
• Instantaneous Velocity: $v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

Acceleration ($\mathbf{a}$): The rate of change of velocity.

• Average Acceleration: $\bar{a} = \frac{\Delta v}{\Delta t}$
• Instantaneous Acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Equations of Motion (Constant Acceleration)

For motion with constant acceleration $a$, the following kinematic equations apply:

1. $v_f = v_i + a t$
2. $\Delta x = v_i t + \frac{1}{2} a t^2$
3. $v_f^2 = v_i^2 + 2 a \Delta x$
4. $\Delta x = \frac{1}{2} (v_i + v_f) t$

Free Fall

Free fall is a special case of 1D motion where the only force acting on an object is gravity. Near the Earth's surface, the acceleration due to gravity is approximately constant: $g \approx 9.81 \text{ m/s}^2$ The acceleration vector is directed downwards. If we choose the upward direction as positive ($+y$), then $a_y = -g$.

The kinematic equations for free fall become:

1. $v_y = v_{yi} - g t$
2. $\Delta y = v_{yi} t - \frac{1}{2} g t^2$
3. $v_y^2 = v_{yi}^2 - 2 g \Delta y$

Motion in 2D

Projectile Motion

Projectile motion is the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. The motion can be analyzed as two independent 1D motions:

• Horizontal Motion: Constant velocity ($a_x = 0$).
• Vertical Motion: Constant acceleration ($a_y = -g$).

Equations:

• Horizontal: $x = x_i + v_{xi} t$ $v_x = v_{xi} = \text{constant}$
• Vertical: $y = y_i + v_{yi} t - \frac{1}{2} g t^2$ $v_y = v_{yi} - g t$

Uniform Circular Motion

An object moving in a circle at constant speed is in uniform circular motion. Even though speed is constant, velocity is changing direction, so there is an acceleration.

Centripetal Acceleration ($a_c$): Directed toward the center of the circle. $a_c = \frac{v^2}{r}$ where $v$ is the speed and $r$ is the radius.

Period ($T$): Time for one complete revolution. $T = \frac{2 \pi r}{v}$