Kinematics of Particles

Kinematics of Particles

Learning Objectives

Understand the fundamental kinematic variables: position, velocity, and acceleration.
Analyze rectilinear motion under constant acceleration.
Resolve motion into radial/transverse, rectangular, and normal/tangential components.
Apply kinematic principles to solve curvilinear and projectile motion problems.

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 $O$ .

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.

Velocity Equation

The fundamental derivative definition of velocity.

v = \frac{ds}{dt}

Variables

Symbol	Description	Unit
$v$	Velocity	m/s
$s$	Position	m
$t$	Time	s

Acceleration ( $a$ )

The time rate of change of the velocity. It indicates how quickly the velocity is changing.

Acceleration Equations

The fundamental derivative definition of acceleration.

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

Variables

Symbol	Description	Unit
$a$	Acceleration	$m/s^2$
$v$	Velocity	m/s
$s$	Position	m
$t$	Time	s

Fundamental Differential Relation

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

Fundamental Differential Equation of Kinematics

Relation connecting acceleration, displacement, and velocity without time.

a \, ds = v \, dv

Variables

Symbol	Description	Unit
$a$	Acceleration	$m/s^2$
$s$	Position	m
$v$	Velocity	m/s

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 $a_y = -g \approx -9.81 \, \text{m/s}^2$ or $-32.2 \, \text{ft/s}^2$ .

Constant Acceleration Equations

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

v = v_0 + a_c t

Variables

Symbol	Description	Unit
$v$	Final velocity	m/s
$v_0$	Initial velocity at t=0	m/s
$a_c$	Constant acceleration	$m/s^2$
$t$	Time elapsed	s

Position as a Function of Time

Analytical formula for position as a function of time with uniform acceleration.

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

Variables

Symbol	Description	Unit
$s$	Final position	m
$s_0$	Initial position	m
$v_0$	Initial velocity	m/s
$a_c$	Constant acceleration	$m/s^2$
$t$	Time elapsed	s

Velocity as a Function of Position

Analytical formula for velocity as a function of position with uniform acceleration.

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

Variables

Symbol	Description	Unit
$v$	Final velocity	m/s
$v_0$	Initial velocity	m/s
$a_c$	Constant acceleration	$m/s^2$
$s$	Final position	m
$s_0$	Initial position	m

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.

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 ( $r$ , $\theta$ )

The radial and transverse components of velocity.

v_r = \dot{r} \quad \text{and} \quad v_\theta = r \dot{\theta}

Variables

Symbol	Description	Unit
$v_r$	Radial velocity	m/s
$\dot{r}$	Rate of change of radial distance	m/s
$v_\theta$	Transverse velocity	m/s
$r$	Radial distance	m
$\dot{\theta}$	Rate of change of angle (angular velocity)	rad/s

Acceleration Components ( $r$ , $\theta$ )

The radial and transverse components of acceleration.

a_r = \ddot{r} - r \dot{\theta}^2 \quad \text{and} \quad a_\theta = r \ddot{\theta} + 2 \dot{r} \dot{\theta}

Variables

Symbol	Description	Unit
$a_r$	Radial acceleration	$m/s^2$
$\ddot{r}$	Second derivative of radial distance	$m/s^2$
$a_\theta$	Transverse acceleration	$m/s^2$
$\ddot{\theta}$	Second derivative of angle (angular acceleration)	$rad/s^2$
$r$	Radial distance	m
$\dot{r}$	First derivative of radial distance	m/s
$\dot{\theta}$	First derivative of angle	rad/s

Interactive Simulation

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

Radial and Transverse Kinematics Visualizer

Controls

Radial Position (r)120.0 m

Angle (θ)45.0°

Radial Velocity (

\dot{r} = v_r

)20.0 m/s

Angular Velocity (

\dot{\theta} = \omega

)0.50 rad/s

Mathematical Kinematics

Velocity components:

v_r = \dot{r} = 20.0\text{ m/s}

v_\theta = r\dot{\theta} = (120)(0.50) = 60.0\text{ m/s}

v = \sqrt{v_r^2 + v_\theta^2} = 63.2\text{ m/s}

Acceleration (

\ddot{r}=0, \\ddot{\\theta}=0

a_r = -r\dot{\theta}^2 = -(120)(0.50)^2 = -30.0\text{ m/s}^2

a_\theta = 2\dot{r}\dot{\theta} = 2(20)(0.50) = 20.0\text{ m/s}^2

a = \sqrt{a_r^2 + a_\theta^2} = 36.1\text{ m/s}^2

v_r

(Radial Velocity)

v_\theta

(Transverse Velocity)

v

(Total Velocity)

a_r

(Radial Accel.)

a_\theta

(Transverse Accel.)

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.

Horizontal Motion ( $x$ ): Since there are no horizontal forces, acceleration $a_x = 0$ . Velocity remains constant.

Projectile Motion - Horizontal

Equations for initial horizontal velocity and horizontal position.

(v_x)_0 = v_0 \cos \theta \quad \text{and} \quad x = x_0 + (v_x)_0 t

Variables

Symbol	Description	Unit
$(v_x)_0$	Initial horizontal velocity	m/s
$v_0$	Initial overall velocity	m/s
$\theta$	Launch angle	rad
$x$	Horizontal position at time t	m
$x_0$	Initial horizontal position	m
$t$	Time	s

Vertical Motion ( $y$ ): Gravity provides constant downward acceleration, $a_y = -g$ .

Projectile Motion - Vertical

Equations for vertical velocity and position under constant gravity.

\begin{aligned} (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) \end{aligned}

Variables

Symbol	Description	Unit
$(v_y)_0$	Initial vertical velocity	m/s
$v_0$	Initial overall velocity	m/s
$\theta$	Launch angle	rad
$v_y$	Vertical velocity at time t	m/s
$g$	Acceleration due to gravity	$m/s^2$
$t$	Time	s
$y$	Vertical position at time t	m
$y_0$	Initial vertical position	m

Interactive Simulation

Interact with the simulation below to observe projectile motion parameters.

Projectile Motion Simulator

Launch Options

Velocity (

v_0

)30 m/s

Launch Angle (

\theta

)45°

Cliff Height (

y_0

)20 m

Air Resistance (Quadratic Drag)

Scrub Time0.00s / 5.13s

Motion State & Math

Max Height:43.04 m

Range:108.82 m

Flight Time:5.13 s

Live Particle State:

x: 0.0 m

y: 20.0 m

v_x: 21.2 m/s

v_y: 21.2 m/s

Speed: 30.0 m/s

Governing Physics:

a_x = 0, \quad a_y = -g

y(t) = y_0 + v_{0y}t - \frac{1}{2}gt^2

v_x (Horizontal Vel.)

v_y (Vertical Vel.)

v (Total Vel.)

g (Gravity)

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.

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

Tangential Acceleration

Acceleration tangent to the path, causing a change in speed.

a_t = \frac{dv}{dt}

Variables

Symbol	Description	Unit
$a_t$	Tangential acceleration	$m/s^2$
$v$	Velocity (speed)	m/s
$t$	Time	s

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.

Normal Acceleration

Acceleration normal to the path, directed towards the center of curvature.

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

Variables

Symbol	Description	Unit
$a_n$	Normal acceleration	$m/s^2$
$v$	Velocity (speed)	m/s
$\rho$	Radius of curvature	m

The magnitude of the total acceleration is determined from its normal and tangential components.

Total Acceleration Magnitude

The vector magnitude of total acceleration.

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

Variables

Symbol	Description	Unit
$a$	Total acceleration magnitude	$m/s^2$
$a_t$	Tangential acceleration	$m/s^2$
$a_n$	Normal acceleration	$m/s^2$

Constant Speed on a Curve

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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Learning Objectives

Rectilinear Motion

Position (sss)

Displacement (Δs\Delta sΔs)

Velocity (vvv)

Velocity Equation

Acceleration (aaa)

Acceleration Equations

Fundamental Differential Relation

Fundamental Differential Equation of Kinematics

Motion with Constant Acceleration

Constant Acceleration Equations

Position as a Function of Time

Velocity as a Function of Position

Radial and Transverse Coordinates (rrr, θ\thetaθ)

Velocity Components (rrr, θ\thetaθ)

Acceleration Components (rrr, θ\thetaθ)

Interactive Simulation

Controls

Mathematical Kinematics

Curvilinear Motion

Rectangular Components (x, y, z)

Projectile Motion

Projectile Motion - Horizontal

Projectile Motion - Vertical

Interactive Simulation

Projectile Motion Simulator

Launch Options

Motion State & Math

Normal and Tangential Components (n-t)

Tangential Acceleration

Normal Acceleration

Total Acceleration Magnitude

Constant Speed on a Curve

Position ( $s$ )

Displacement ( $\Delta s$ )

Velocity ( $v$ )

Acceleration ( $a$ )

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

Velocity Components ( $r$ , $\theta$ )

Acceleration Components ( $r$ , $\theta$ )