Understanding the fundamental principles of motion is crucial in physics, especially when dealing with objects moving under constant acceleration. Equations of motion are mathematical formulas that relate the displacement, velocity (initial and final), acceleration, and time of an object in motion. These equations are derived for situations where the acceleration remains constant.

Motion Equations Overview

For an object moving with uniform or constant acceleration, there are three primary equations of motion that are widely used in kinematics. These standard forms are:

• First Equation of Motion: \(v = u + at\)
• Second Equation of Motion: \(s = ut + \frac{1}{2}at^2\)
• Third Equation of Motion: \(v^2 = u^2 + 2as\) (which can be rearranged as \(2as = v^2 - u^2\))

Analyzing Statement I: \(u = v + at\)

Let's examine the first statement given: \(u = v + at\).

The standard first equation of motion relates final velocity (\(v\)), initial velocity (\(u\)), acceleration (\(a\)), and time (\(t\)) as:

\(v = u + at\)

If we rearrange this standard equation to solve for initial velocity (\(u\)), we get:

\(u = v - at\)

Comparing this correct rearrangement \(u = v - at\) with the given statement \(u = v + at\), we can see that the given statement has an incorrect sign for the acceleration term. The initial velocity \(u\) should decrease from final velocity \(v\) if there is a positive acceleration \(a\) over time \(t\). Therefore, statement I, in the exact form \(u = v + at\), is not a correct standard equation of motion.

Analyzing Statement II: \(2as = v^2 - u^2\)

Now let's consider the second statement: \(2as = v^2 - u^2\).

The standard third equation of motion, often written as \(v^2 = u^2 + 2as\), relates final velocity (\(v\)), initial velocity (\(u\)), acceleration (\(a\)), and displacement (\(s\)).

If we rearrange the standard third equation, \(v^2 = u^2 + 2as\), by subtracting \(u^2\) from both sides, we get:

\(v^2 - u^2 = 2as\)

This is exactly the form presented in statement II. Hence, statement II, \(2as = v^2 - u^2\), is a correct and standard equation of motion used in kinematics.

Conclusion on Equations of Motion

Based on our detailed analysis of both statements:

• Statement I (\(u = v + at\)) is not a correctly formulated equation of motion, as its form deviates from the standard first equation (\(v = u + at\)) or its correct rearrangement (\(u = v - at\)).
• Statement II (\(2as = v^2 - u^2\)) is a correctly formulated equation of motion, directly representing a rearrangement of the standard third equation of motion (\(v^2 = u^2 + 2as\)).

Therefore, only statement II is an equation of motion from the given options.