Consider a body moving with initial velocity u. Let its velocity change to \(\mathrm{v}\), in time ' \(\mathrm{t}^{\prime}\). Then, the change in velocity is \(=\mathrm{v}-\mathrm{u}\). The change in velocity per unit time \(=\frac{\mathrm{v}-\mathrm{u}}{\mathrm{t}}\) By definition, the change in velocity per unit time is acceleration, a. Thus, \(\mathrm{a}=\frac{\mathrm{v}-\mathrm{u}}{\mathrm{t}}\) or \(\mathrm{v}-\mathrm{u}=\) at \(\mathrm{v}=\mathrm{u}+\mathrm{at}\)

Understanding the initial velocity is fundamental to studying motion in physics. It is the speed at which an object starts its journey before any external forces act upon it to change this speed. Imagine you're tossing a ball into the air; the initial velocity would be the speed of the ball as it leaves your hand.

Represented mathematically by the letter 'u', initial velocity is a vector quantity, meaning it has both magnitude (speed) and direction. In equations, it acts as the starting point for calculating changes in motion, including how fast something will be moving at a later time given certain conditions. It's essential to know the initial velocity to predict the object's future position and velocity.

The term 'final velocity' refers to the speed of an object at the end of the considered time period and is denoted by 'v' in equations. If you watch a car accelerate down the street, its final velocity is how fast it's going at the moment you stop timing.

This concept is crucial because it can tell us whether an object has accelerated, decelerated, or maintained a constant speed over time. Final velocity is calculated based on the initial velocity, the acceleration (or deceleration) that the object has undergone, and the amount of time over which this change occurred. It helps in predicting the object's future motion or in determining the impact of different forces that the object has encountered during its movement.

The 'change in velocity' is defined as the difference between the final velocity and the initial velocity. It is a vector quantity that provides information about the magnitude and direction of the acceleration.

In mathematical terms, it is expressed as \( \Delta v = v - u \), showing how much an object's speed and direction of travel have altered over a given time frame. For example, if a cyclist starts at a speed of 5 meters per second and then accelerates to 15 meters per second, the change in velocity is 10 meters per second. This change is critical in calculating other aspects of motion, such as displacement and kinetic energy, making it a foundational concept in kinematics.

In physics, 'unit time' is a standard duration used for measuring how quickly or slowly changes occur. It standardizes the rate of change, making it easier to compare and contrast different scenarios. When discussing acceleration, unit time is usually in seconds, represented by 't' in equations.

By looking at changes per unit time, we can understand rates of acceleration or deceleration. This helps in quantifying motion in an easily interpretable way. For instance, when we discuss acceleration as meters per second squared (m/s\(^2\)), we're saying that for each second, the velocity changes by a certain amount of meters per second. Thus, unit time serves as a fundamental measuring stick for time-dependent changes in physical quantities.