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

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

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

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:

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

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 $v^2 = v_0^2 + 2a(x - x_0)$.

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.

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).

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 ($t$).

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:

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

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

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.

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:

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 ($\vec{v}_{M/E}$):

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:

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.