First Equation of Motion – Derivation

he first equation of motion describes the relationship between the final velocity (

$v$), initial velocity ($u$), acceleration ($a$), and time ($t$). It is expressed mathematically as:

$$v = u + at$$

This equation is fundamental in kinematics, particularly for objects moving with uniform acceleration. Let’s delve into its derivation step-by-step.

Understanding Acceleration

Acceleration is defined as the rate of change of velocity over time. Mathematically, this can be expressed as:

$$a = \frac{v - u}{t}$$

Here:

• $a$ is the acceleration,
• $v$ is the final velocity,
• $u$ is the initial velocity,
• $t$ is the time over which the acceleration occurs.

Rearranging this equation allows us to express the final velocity in terms of the other variables:

$$v - u = at$$

By adding $u$ to both sides, we arrive at:

$$v = u + at$$

This simple rearrangement leads us to the first equation of motion, indicating that the final velocity is the initial velocity plus the product of acceleration and time.

Conceptual Explanation

To understand this equation conceptually, consider an object that starts moving with an initial velocity $u$ and experiences a constant acceleration $a$ for a duration of time $t$.

1. Initial Velocity: The object begins its motion with a certain speed, $u$. This could be a stationary object starting to move or one already in motion.

2. Acceleration: Acceleration ($a$) refers to how quickly the object's velocity changes. If $a$ is positive, the object is speeding up; if negative, it is slowing down.

3. Time: The time interval ($t$) over which this acceleration occurs is crucial. The longer the time under acceleration, the greater the change in velocity.

Practical Example

Imagine a car at a stoplight. When the light turns green, it accelerates from rest ($u = 0$) at a rate of $2 \, \text{m/s}^2$ for $5$ seconds. Using the first equation of motion, we can determine the final velocity:

• Initial velocity ($u$) = 0 m/s
• Acceleration ($a$) = 2 m/s²
• Time ($t$) = 5 s

Substituting these values into the equation:

$$v = 0 + (2 \, \text{m/s}^2)(5 \, \text{s}) = 10 \, \text{m/s}$$

Thus, the car reaches a final velocity of $10 \, \text{m/s}$ after accelerating for $5$ seconds.

Graphical Interpretation

Graphically, the first equation can be understood in the context of a velocity-time graph. The slope of this graph represents acceleration. If you plot initial velocity on the y-axis and time on the x-axis, the linear relationship shows that the change in velocity is proportional to time, confirming the equation $v = u + at$.

The first equation of motion encapsulates a fundamental aspect of motion: how an object’s velocity changes over time when subjected to constant acceleration. It provides a concise and effective means of predicting future velocities in a variety of physical contexts, from everyday scenarios to more complex systems in physics and engineering. Understanding this equation is crucial for solving a wide range of problems involving motion.