Kinematics of Particles

Kinematics of Particles

Kinematics is the branch of mechanics that deals exclusively with the geometric aspects of motion. It details the trajectory of a particle over time without any reference to the mass of the particle or the forces that cause the motion. In this section, we treat bodies as particles, which means we assume all mass is concentrated at a single point, allowing us to neglect dimensions, shape, and rotational effects.

Rectilinear Motion

Rectilinear motion describes a particle moving along a single straight line. The position of the particle at any given instant is defined by a single coordinate, typically denoted as

s

(or

x

y

), measured from a fixed origin

O

Fundamental Kinematic Variables

Position ( $s$ ): The linear coordinate that locates the particle on the straight line at time

t

Displacement ( $\Delta s$ ): The change in position over a time interval

\Delta t

\Delta s = s_2 - s_1

. It is a vector quantity (though simplified to a scalar with a sign in 1D).

Velocity ( $v$ ): The time rate of change of the position. It indicates both the speed and direction of motion.

v = \frac{ds}{dt}

Acceleration ( $a$ ): The time rate of change of the velocity. It indicates how quickly the velocity is changing.

a = \frac{dv}{dt} = \frac{d^2s}{dt^2}

Important

By applying the chain rule, we can eliminate time

dt

between the velocity and acceleration definitions to obtain a fundamental differential relation connecting displacement, velocity, and acceleration:

a = \frac{dv}{dt} = \frac{dv}{ds} \cdot \frac{ds}{dt} = v \frac{dv}{ds}

a \, ds = v \, dv

This equation is highly useful when time is not an explicitly given variable in a problem.

Motion with Constant Acceleration

When a particle moves with a uniform, constant acceleration (

a_c

), the fundamental differential equations can be integrated analytically. This scenario frequently occurs, for example, for objects in free fall near the Earth's surface where

a_y = -g \approx -9.81 \, \text{m/s}^2

-32.2 \, \text{ft/s}^2

Constant Acceleration Equations

Analytical formulas for velocity as a function of time with uniform acceleration.

$$
v = v_0 + a_c t
$$

Additional Constant Acceleration Formulas

Position as a Function of Time:
$s = s_0 + v_0 t + \frac{1}{2} a_c t^2$
Velocity as a Function of Position:
$v^2 = v_0^2 + 2 a_c (s - s_0)$

r

When the position of a particle is defined using polar coordinates (

r

\theta

), it is often necessary to resolve the velocity and acceleration into radial (along

r

) and transverse (perpendicular to

r

) components. This is especially useful in orbital mechanics and problems involving rotation about a central point.

Velocity and Acceleration ( $r$ , $\theta$ )

The position vector is given by

\mathbf{r} = r \mathbf{u}_r

, where

\mathbf{u}_r

is the unit vector in the radial direction.

Velocity Components:

Radial Velocity ( $v_r$ ): The rate at which the distance from the origin changes. $v_r = \dot{r}$
Transverse Velocity ( $v_\theta$ ): The rate at which the angle changes, determining the tangential speed. $v_\theta = r \dot{\theta}$

Acceleration Components:

Radial Acceleration ( $a_r$ ): $a_r = \ddot{r} - r \dot{\theta}^2$
Transverse Acceleration ( $a_\theta$ ): $a_\theta = r \ddot{\theta} + 2 \dot{r} \dot{\theta}$

Interact with the simulation below to explore radial and transverse components of motion.

Radial and Transverse Velocity Components

Radius (r)150 m

Angle (θ)45°

Radial Velocity (v_r)40 m/s

Transverse Velocity (v_θ)60 m/s

Total Velocity:

|\mathbf{v}| = \sqrt{v_r^2 + v_\theta^2} = 72.1 \text{ m/s}

Curvilinear Motion

Curvilinear motion occurs when a particle travels along a curved path in two or three dimensions. Because the direction of motion is constantly changing, the velocity vector changes direction, meaning the particle experiences acceleration even if its speed is constant.

Rectangular Components (x, y, z)

When the path is easily described in Cartesian coordinates, the motion can be resolved into independent horizontal (

x

) and vertical (

y

) components. The position vector is

\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}

. Velocity is

\mathbf{v} = \dot{x}\mathbf{i} + \dot{y}\mathbf{j} + \dot{z}\mathbf{k}

and acceleration is

\mathbf{a} = \ddot{x}\mathbf{i} + \ddot{y}\mathbf{j} + \ddot{z}\mathbf{k}

Projectile Motion

Projectile motion is a classic 2D curvilinear motion where a particle is launched and moves under the sole influence of gravity (neglecting aerodynamic drag). It is analyzed as two independent rectilinear motions.

Projectile Motion Principles

Horizontal Motion ( $x$ ): Since there are no horizontal forces, acceleration

a_x = 0

. Velocity remains constant.

(v_x)_0 = v_0 \cos \theta

x = x_0 + (v_x)_0 t

Vertical Motion ( $y$ ): Gravity provides constant downward acceleration,

a_y = -g

(v_y)_0 = v_0 \sin \theta

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)

Interact with the simulation below to observe projectile motion parameters.

Projectile Motion Simulator

Velocity (

v_0

)30 m/s

Angle (

\theta

)45°

Max Height:22.94 m

Range:91.74 m

Flight Time:4.32 s

Current Time:0.00 s

Normal and Tangential Components (n-t)

When the actual path of a particle is known, it is often most convenient to define motion using a path-fixed coordinate system with axes normal (

n

) and tangential (

t

) to the path.

Normal and Tangential Acceleration

Tangential Acceleration ( $a_t$ ): Represents the rate of change of the magnitude of the velocity (speed). It acts tangent to the path.

a_t = \frac{dv}{dt}

Normal Acceleration ( $a_n$ ): Represents the rate of change of the direction of the velocity. It always acts normal to the path, directed towards the center of curvature.

a_n = \frac{v^2}{\rho}

Where

\rho

is the radius of curvature at that point on the path. The magnitude of the total acceleration is

a = \sqrt{a_t^2 + a_n^2}

Important

For a particle moving along a curve, normal acceleration (

a_n

) is never zero unless the velocity is zero. Even if a car is traveling at a constant speed (

a_t = 0

) around a curve, it still experiences normal acceleration due to the change in direction.

Key Takeaways

Rectilinear Kinematics: Uses the fundamental equations $v=ds/dt$ , $a=dv/dt$ , and $a\,ds=v\,dv$ .
Constant Acceleration: Allows explicit formulas mapping position, velocity, and time ( $v=v_0+at$ , etc.).
Projectile Motion: A 2D motion superposition of constant horizontal velocity and constant vertical acceleration (gravity).
Normal Acceleration ( $a_n = v^2/\rho$ ) accounts for change in direction, while Tangential Acceleration ( $a_t = dv/dt$ ) accounts for change in speed.
The radius of curvature $\rho$ controls direction changes along a curved path.

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Rectilinear Motion

Fundamental Kinematic Variables

Important

Motion with Constant Acceleration

Constant Acceleration Equations

Additional Constant Acceleration Formulas

Radial and Transverse Coordinates (rrr, θ\thetaθ)

Velocity and Acceleration (rrr, θ\thetaθ)

Radial and Transverse Velocity Components

Curvilinear Motion

Rectangular Components (x, y, z)

Projectile Motion

Projectile Motion Principles

Projectile Motion Simulator

Normal and Tangential Components (n-t)

Normal and Tangential Acceleration

Important

Radial and Transverse Coordinates ( $r$ , $\theta$ )

Velocity and Acceleration ( $r$ , $\theta$ )