Kinematics of Particles

Kinematics is the branch of mechanics that deals with the geometry of motion without reference to the forces that cause the motion. In this section, we treat bodies as particles, meaning we neglect their dimensions and rotation.

Rectilinear Motion

Rectilinear motion describes a particle moving along a straight line. The position of the particle is defined by a single coordinate, $s$ , measured from a fixed origin $O$ .

Fundamental Variables

Position ( $s$ ): The location of the particle at any instant $t$ .
Velocity ( $v$ ): The time rate of change of position. $v = \frac{ds}{dt}$
Acceleration ( $a$ ): The time rate of change of velocity. $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$
Differential Relation: Eliminating $dt$ between the velocity and acceleration equations gives: $a \, ds = v \, dv$

Motion with Constant Acceleration

If acceleration $a$ is constant (e.g., gravity), we can integrate the fundamental equations to obtain:

v = v_0 + a_c t

s = s_0 + v_0 t + \frac{1}{2} a_c t^2

v^2 = v_0^2 + 2 a_c (s - s_0)

Where $v_0$ and $s_0$ are the initial velocity and position at $t=0$ .

Example: Braking Car

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Curvilinear Motion: Projectile Motion

Curvilinear motion occurs when a particle moves along a curved path. Projectile motion is a special case where a particle is launched into the air and moves under the influence of gravity alone (neglecting air resistance).

Equations of Projectile Motion

The motion is analyzed by separating it into independent horizontal ( $x$ ) and vertical ( $y$ ) components. Assume $a_x = 0$ and $a_y = -g$ .

Horizontal Motion: $v_x = (v_x)_0$ $x = x_0 + (v_x)_0 t$

Vertical Motion: $v_y = (v_y)_0 - gt$ $y = y_0 + (v_y)_0 t - \frac{1}{2} gt^2$ $v_y^2 = (v_y)_0^2 - 2g(y - y_0)$

Example: Projectile Range

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Normal and Tangential Components

When the path of motion is known, it is often convenient to describe the motion using $n$ (normal) and $t$ (tangential) coordinates.

Tangential Axis ( $t$ ): Tangent to the curve, positive in the direction of motion.
Normal Axis ( $n$ ): Perpendicular to the tangent, pointing toward the center of curvature.

n-t Acceleration Components

Tangential Acceleration ( $a_t$ ): Changes the magnitude of velocity (speed). $a_t = \frac{dv}{dt}$

Normal Acceleration ( $a_n$ ): Changes the direction of velocity. $a_n = \frac{v^2}{\rho}$

Where $\rho$ is the radius of curvature. The total magnitude of acceleration is $a = \sqrt{a_t^2 + a_n^2}$ .

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