Kinematics

Kinematics

$Learning Objectives$

Define and calculate displacement, velocity, and acceleration with civil engineering context.
Understand the difference between average and instantaneous kinematic quantities.
Derive and apply the kinematic equations for 1D motion with constant acceleration.
Analyze 2D projectile motion by separating it into independent horizontal and vertical components.
Calculate tangential and centripetal acceleration in circular motion for transportation design.
Solve relative motion problems in 1D and 2D using vector addition.

Kinematics is the branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without considering the forces that cause them to move. It is essentially the "geometry of motion." Before we can understand why objects move (Dynamics), we must first master how to describe that motion. Kinematics provides the vocabulary and equations necessary to track an object's position, speed, and acceleration over time. This is essential for traffic engineering, robotics, and aerospace trajectories.

Key Kinematic Quantities

$Key Kinematic Quantities Concepts$

To describe motion, we use a specific set of parameters. All of these (except time and distance) are vector quantities.

$\Delta x$

The change in position of an object. It is a vector pointing from the initial position to the final position.

\Delta \vec{x} = \vec{x}_f - \vec{x}_i

Note: Displacement is not the same as the total distance traveled, which is a scalar representing the actual path length.

$Kinematics relationship map$

Position, velocity, and acceleration form a calculus chain; displacement and velocity change are accumulated areas.

Position, velocity, and acceleration linked by differentiation and integration, with displacement and velocity change shown as integrals. — The kinematics chain
Differentiation moves from position to velocity to acceleration; integration reverses the chain and accumulates signed changes.

Average vs Instantaneous

$Average vs Instantaneous Concepts$

In kinematics, it is critical to distinguish between average values over a time interval and instantaneous values at a specific moment.

$\vec{v}_{avg}$

The total displacement divided by the time interval during which the displacement occurred. It points in the same direction as the displacement.

\vec{v}_{avg} = \frac{\Delta \vec{x}}{\Delta t}

Contrast this with average speed, which is total distance divided by total time (a scalar).

$\vec{v}(t)$

The velocity of an object at a specific instant in time. Mathematically, it is the limit of the average velocity as the time interval approaches zero. It is the derivative of position with respect to time.

\vec{v}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}

$\vec{a}$

The rate of change of velocity with respect to time. An object is accelerating if its speed changes, its direction changes, or both.

\vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t}

\vec{a}(t) = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{x}}{dt^2}

Graphical Analysis of Motion

$Position-Time Graph (x vs t)$

A graph where the slope of the tangent line at any point represents the instantaneous velocity. A constant slope indicates constant velocity. A curve indicates changing velocity (acceleration).

$Velocity-Time Graph (v vs t)$

A graph where the slope of the tangent line represents instantaneous acceleration. The area under the curve between two times represents the displacement.

$Acceleration-Time Graph (a vs t)$

A graph where the area under the curve represents the change in velocity.

$Graphical Analysis of Motion Concepts$

Graphs are powerful tools for visualizing kinematics. The slope and area under curves provide vital information regarding position, velocity, and acceleration over time.

$Motion-graph interpretation$

Slope and signed area connect the three standard motion graphs. These interpretations remain valid beyond constant-acceleration cases when the curves are handled correctly.

Position-time, velocity-time, and acceleration-time graphs annotated with slope and signed-area relationships. — Motion graph relationships
Slope of x-t gives velocity, slope of v-t gives acceleration, area under v-t gives displacement, and area under a-t gives velocity change.

Deriving the Kinematic Equations (Calculus Approach)

$Deriving the Kinematic Equations (Calculus Approach) Concepts$

For engineering mechanics, it is essential to understand that kinematic equations are derived using fundamental calculus operations on the definitions of velocity and acceleration.

Given $a = dv/dt$ , we can integrate with respect to time to find velocity. If acceleration $a$ is constant:

Velocity from Constant Acceleration

Integrates constant acceleration over time to find the velocity equation.

\int_{v_0}^{v} dv = \int_{0}^{t} a \, dt \implies v = v_0 + at

Variables

Symbol	Description	Unit
$v$	Final velocity	m/s
$v_0$	Initial velocity	m/s
$a$	Constant acceleration	$m/s^2$
$t$	Time interval	s

$Deriving Position from Velocity$

Given $v = dx/dt$ , we substitute the velocity equation and integrate again to find position:

Position from Constant Acceleration

Integrates velocity over time to find the position equation under constant acceleration.

\int_{x_0}^{x} dx = \int_{0}^{t} (v_0 + at) \, dt \implies x = x_0 + v_0 t + \frac{1}{2}at^2

Variables

Symbol	Description	Unit
$x$	Final position	m
$x_0$	Initial position	m
$v_0$	Initial velocity	m/s
$a$	Constant acceleration	$m/s^2$
$t$	Time interval	s

$Time-Independent Derivation$

If we need a relationship independent of time, we use the chain rule: $a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx}$ . Separating variables and integrating yields $v^2 = v_0^2 + 2a(x - x_0)$ .

1D Motion with Constant Acceleration

$1D Motion with Constant Acceleration Concepts$

When an object's acceleration is constant (like a car braking uniformly, or an object in free fall near Earth's surface), its motion can be completely described by four foundational equations, often called the "Big Four" kinematic equations. As shown above, these are derived directly from calculus.

Kinematic Equation 1: Velocity as a Function of Time

Calculates final velocity given initial velocity, constant acceleration, and time.

v_f = v_i + at

Variables

Symbol	Description	Unit
$v_f$	Final velocity	m/s
$v_i$	Initial velocity	m/s
$a$	Constant acceleration	$m/s^2$
$t$	Time interval	s

Kinematic Equation 2: Displacement as a Function of Time

Calculates displacement given initial velocity, time, and constant acceleration.

\Delta x = v_i t + \frac{1}{2}at^2

Variables

Symbol	Description	Unit
$\Delta x$	Displacement	m
$v_i$	Initial velocity	m/s
$a$	Constant acceleration	$m/s^2$
$t$	Time interval	s

Kinematic Equation 3: Velocity as a Function of Displacement

Calculates final velocity without knowing the time interval.

v_f^2 = v_i^2 + 2a\Delta x

Variables

Symbol	Description	Unit
$v_f$	Final velocity	m/s
$v_i$	Initial velocity	m/s
$a$	Constant acceleration	$m/s^2$
$\Delta x$	Displacement	m

Kinematic Equation 4: Displacement using Average Velocity

Calculates displacement using initial and final velocity without knowing acceleration.

\Delta x = \left(\frac{v_i + v_f}{2}\right)t

Variables

Symbol	Description	Unit
$\Delta x$	Displacement	m
$v_i$	Initial velocity	m/s
$v_f$	Final velocity	m/s
$t$	Time interval	s

$Interactive Simulation$

Use the graph simulator to see how changing initial velocity, acceleration, and time reshapes the position-time and velocity-time curves.

Select Active Graph

6.0 m/s

1.5 m/s²

8.0 s

4.0 s

Governing Formulas

Position & Velocity

x(t) = v_0 t + \frac{1}{2}a t^2, \quad v(t) = v_0 + a t

Calculus Theorem

v(t) = \frac{dx}{dt}, \quad x(t) = \int_{0}^{t} v(\tau) d\tau \quad (\text{Area})

Position (x)

36.00 m

Velocity (v)

12.00 m/s

Acceleration

1.50 m/s²

Free Fall

$Free Fall Concepts$

A classic example of constant 1D acceleration is an object falling under the sole influence of gravity near the Earth's surface. In this case, the acceleration is constant and directed downwards: $a = -g = -9.81 \text{ m/s}^2$ (assuming the positive y-axis points upward).

Note: In true free fall, the mass, size, and shape of the object do not affect its acceleration (neglecting air resistance).

2D Motion: Projectile Motion

$2D Motion: Projectile Motion Concepts$

When an object is launched into the air and moves in two dimensions under the influence of gravity alone, it is in projectile motion. A key assumption is that air resistance is negligible.

The fundamental principle for solving projectile motion problems is the independence of motion: the horizontal ( $x$ ) and vertical ( $y$ ) motions are completely independent of each other, except that they share the same elapsed time ( $t$ ).

$Projectile Motion Principles$

Horizontal Motion (x-axis):

Because gravity acts purely vertically, there is no horizontal acceleration.

Acceleration is zero ( $a_x = 0$ ).
Velocity is constant ( $v_x = v_{ix} = \text{constant}$ ).
Equation: $\Delta x = v_{ix} t$

Vertical Motion (y-axis):

The vertical motion is identical to 1D free fall.

Acceleration is constant gravity ( $a_y = -g$ ).
Velocity changes continuously.
Equations: The standard 1D kinematic equations apply, substituting $y$ for $x$ and $-g$ for $a$ .

$Initial Velocity Components$

If a projectile is launched from the origin with an initial velocity $v_0$ at an angle $\theta$ above the horizontal, we must resolve the initial velocity into components:

Initial Horizontal Velocity

The initial horizontal component of the projectile's velocity.

v_{ix} = v_0 \cos\theta

Variables

Symbol	Description	Unit
$v_{ix}$	Initial horizontal velocity	m/s
$v_0$	Initial launch speed	m/s
$\theta$	Launch angle relative to horizontal	deg/rad

Initial Vertical Velocity

The initial vertical component of the projectile's velocity.

v_{iy} = v_0 \sin\theta

Variables

Symbol	Description	Unit
$v_{iy}$	Initial vertical velocity	m/s
$v_0$	Initial launch speed	m/s
$\theta$	Launch angle relative to horizontal	deg/rad

$Interactive Simulation$

Adjust launch conditions below to connect horizontal constant velocity with vertical free-fall acceleration.

Planetary Environment

25 m/s

40 deg

4.0 m

0.40

Planetary Ballistics Equations

Horizontal Motion (No Air Resistance):

x(t) = (v_0 \cos\theta) \cdot t

Vertical Kinematics with Planetary Gravity:

y(t) = h_0 + (v_0 \sin\theta) \cdot t - \frac{1}{2}g t^2

Gravity (Earth): 9.81 m/s²

Horizontal Range (d)

67.19 m

Max Vertical Height

17.16 m

Flight Duration (t)

1.40 / 3.51 s

Key Projectile Characteristics

$Maximum Height$

Occurs when the vertical velocity $v_y = 0$ .

$Time of Flight$

The total time the projectile is in the air. For a launch and landing at the same elevation, it is twice the time to reach maximum height.

$R$

The horizontal distance covered. For a launch and landing at the same elevation, $R = \frac{v_0^2 \sin(2\theta)}{g}$ . Maximum range is achieved at $\theta = 45^\circ$ .

$Key Projectile Characteristics Concepts$

Understanding key projectile characteristics such as maximum height, time of flight, and range is crucial for analyzing 2D motion under gravity.

Circular Motion Acceleration

$Circular Motion Acceleration Concepts$

When an object moves along a curved path, its acceleration can be broken into two orthogonal components: tangential and normal (centripetal) acceleration.

Tangential Acceleration

The component of acceleration parallel to the velocity vector, changing the speed.

a_t = \frac{dv}{dt}

Variables

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

Centripetal (Normal) Acceleration

The component of acceleration perpendicular to velocity, changing the direction towards the center of curvature.

a_c = \frac{v^2}{r}

Variables

Symbol	Description	Unit
$a_c$	Centripetal acceleration	$m/s^2$
$v$	Magnitude of velocity (speed)	m/s
$r$	Radius of curvature	m

$Uniform Circular Motion$

In uniform circular motion (constant speed), $a_t = 0$ , and the object only experiences centripetal acceleration. In transportation engineering, understanding this centripetal acceleration is critical when designing horizontal curves for highways and railways to ensure vehicles don't slip outward due to insufficient friction or banking.

$Circular-motion kinematics visual$

Even at constant speed, a body on a circular path accelerates because its velocity direction changes continuously.

Object on a circular path with tangent velocity and inward centripetal acceleration vectors. — Velocity and acceleration in circular motion
Velocity is tangent to the path while centripetal acceleration points toward the instantaneous center of curvature.

Relative Motion

$Relative Motion Concepts$

Velocity is not absolute; it depends on the frame of reference of the observer. If you are sitting in a train moving at 50 m/s relative to the ground, your velocity relative to the train is zero, but your velocity relative to an observer on the platform is 50 m/s.

If object A is moving with velocity $\vec{v}_{AE}$ relative to Earth, and object B is moving with velocity $\vec{v}_{BE}$ relative to Earth, the velocity of object A relative to object B ( $\vec{v}_{AB}$ ) is:

1D Relative Velocity

Velocity of object A relative to object B.

\vec{v}_{AB} = \vec{v}_{AE} - \vec{v}_{BE}

Variables

Symbol	Description	Unit
$\vec{v}_{AB}$	Velocity of A relative to B	m/s
$\vec{v}_{AE}$	Velocity of A relative to Earth	m/s
$\vec{v}_{BE}$	Velocity of B relative to Earth	m/s

$Relative Motion Addition$

Alternatively, the velocity of an object relative to a stationary frame ( $\vec{v}_{P/E}$ ) is the sum of the velocity of the object relative to a moving frame ( $\vec{v}_{P/M}$ ) and the velocity of the moving frame relative to the stationary frame ( $\vec{v}_{M/E}$ ):

Relative Velocity Addition

Finding absolute velocity from relative velocity and moving frame velocity.

\vec{v}_{P/E} = \vec{v}_{P/M} + \vec{v}_{M/E}

Variables

Symbol	Description	Unit
$\vec{v}_{P/E}$	Velocity of P relative to Earth	m/s
$\vec{v}_{P/M}$	Velocity of P relative to moving frame M	m/s
$\vec{v}_{M/E}$	Velocity of moving frame M relative to Earth	m/s

Relative Velocity in 2D

$Moving Reference Frames$

While relative motion in 1D is simply vector addition or subtraction along a line, relative velocity in 2D requires vector algebra. This is essential for analyzing the motion of airplanes in crosswinds or boats crossing rivers with currents.

If $\vec{v}_{P/A}$ is the velocity of object P relative to frame A, and $\vec{v}_{A/B}$ is the velocity of frame A relative to frame B, then the velocity of object P relative to frame B is the vector sum:

2D Relative Velocity

Vector addition of relative velocities across different reference frames.

\vec{v}_{P/B} = \vec{v}_{P/A} + \vec{v}_{A/B}

Variables

Symbol	Description	Unit
$\vec{v}_{P/B}$	Velocity of P relative to B	m/s
$\vec{v}_{P/A}$	Velocity of P relative to A	m/s
$\vec{v}_{A/B}$	Velocity of A relative to B	m/s

$Applications$

To solve these problems, engineers decompose the velocity vectors into their $x$ and $y$ components and solve them independently using the standard component method discussed in the previous section.

$Relative-motion vector visual$

Always define the reference frame in the subscripts before combining velocities.

Boat-relative-to-water and current vectors adding to the boat-relative-to-ground resultant. — Relative velocity vector addition
A moving object's absolute velocity equals its velocity relative to a moving frame plus the frame velocity relative to the reference frame.

Key Takeaways

Kinematics describes motion without forces. Key quantities are position, displacement, velocity, and acceleration.
Distinguish between average (interval) and instantaneous (point in time) quantities. Derivatives and integrals link position, velocity, and acceleration.
The Kinematic Equations apply only when acceleration is constant.
For Free Fall, acceleration is constant ( $a = -9.81 \text{ m/s}^2$ ). Mass is irrelevant.
Projectile Motion is analyzed by separating it into independent, constant-velocity horizontal motion and constant-acceleration vertical motion, linked only by time.
Relative velocity requires defining a reference frame and using vector addition/subtraction.

$References$

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Learning Objectives

Key Kinematic Quantities

Key Kinematic Quantities Concepts

Displacement (Δx\Delta xΔx)

Kinematics relationship map

Average vs Instantaneous

Average vs Instantaneous Concepts

Average Velocity (v⃗avg\vec{v}_{avg}vavg​)

Instantaneous Velocity (v⃗(t)\vec{v}(t)v(t))

Acceleration (a⃗\vec{a}a)

Graphical Analysis of Motion

Position-Time Graph (x vs t)

Velocity-Time Graph (v vs t)

Acceleration-Time Graph (a vs t)

Graphical Analysis of Motion Concepts

Motion-graph interpretation

Deriving the Kinematic Equations (Calculus Approach)

Deriving the Kinematic Equations (Calculus Approach) Concepts

Velocity from Constant Acceleration

Deriving Position from Velocity

Position from Constant Acceleration

Time-Independent Derivation

1D Motion with Constant Acceleration

1D Motion with Constant Acceleration Concepts

Kinematic Equation 1: Velocity as a Function of Time

Kinematic Equation 2: Displacement as a Function of Time

Kinematic Equation 3: Velocity as a Function of Displacement

Kinematic Equation 4: Displacement using Average Velocity

Interactive Simulation

Free Fall

Free Fall Concepts

2D Motion: Projectile Motion

2D Motion: Projectile Motion Concepts

Projectile Motion Principles

Initial Velocity Components

Initial Horizontal Velocity

Initial Vertical Velocity

Interactive Simulation

Key Projectile Characteristics

Maximum Height

Time of Flight

Range (RRR)

Key Projectile Characteristics Concepts

Circular Motion Acceleration

Circular Motion Acceleration Concepts

Tangential Acceleration

Centripetal (Normal) Acceleration

Uniform Circular Motion

Circular-motion kinematics visual

Relative Motion

Relative Motion Concepts

1D Relative Velocity

Relative Motion Addition

Relative Velocity Addition

Relative Velocity in 2D

Moving Reference Frames

2D Relative Velocity

Applications

Relative-motion vector visual

References

$Learning Objectives$

$Key Kinematic Quantities Concepts$

$\Delta x$

$Kinematics relationship map$

$Average vs Instantaneous Concepts$

$\vec{v}_{avg}$

$\vec{v}(t)$

$\vec{a}$

$Position-Time Graph (x vs t)$

$Velocity-Time Graph (v vs t)$

$Acceleration-Time Graph (a vs t)$

$Graphical Analysis of Motion Concepts$

$Motion-graph interpretation$

$Deriving the Kinematic Equations (Calculus Approach) Concepts$

$Deriving Position from Velocity$

$Time-Independent Derivation$

$1D Motion with Constant Acceleration Concepts$

$Interactive Simulation$

$Free Fall Concepts$

$2D Motion: Projectile Motion Concepts$

$Projectile Motion Principles$

$Initial Velocity Components$

$Interactive Simulation$

$Maximum Height$

$Time of Flight$

$R$

$Key Projectile Characteristics Concepts$

$Circular Motion Acceleration Concepts$

$Uniform Circular Motion$

$Circular-motion kinematics visual$

$Relative Motion Concepts$

$Relative Motion Addition$

$Moving Reference Frames$

$Applications$

$Relative-motion vector visual$

$References$