Understanding the Equations of Motion

The equations of motion are fundamental concepts in physics, specifically in kinematics, which deals with the motion of objects without considering the forces that cause the motion. These equations relate the initial velocity (u), final velocity (v), acceleration (a), time (t), and displacement (s) of an object moving with constant acceleration along a straight line.

Key Equations of Uniformly Accelerated Motion

For an object moving with constant acceleration 'a', the standard equations of motion are:

1. Velocity-Time Relation: \(v = u + at\)
2. Displacement-Time Relation: \(s = ut + \frac{1}{2}at^2\)
3. Velocity-Displacement Relation: \(v^2 = u^2 + 2as\)

Let's examine each given option and compare it with the standard equations of motion to determine which one is correct.

Analyzing the Given Options

Option 1: \(u = v + at^2\)

This equation is not a standard equation of motion. The term \(at^2\) typically appears in equations related to displacement (s), not directly in the relationship between initial velocity (u) and final velocity (v).

The correct velocity-time relation is \(v = u + at\).

Option 2: \(2s = (v - u)t\)

Let's consider the standard equations. We know that the average velocity is \(\frac{u+v}{2}\). For constant acceleration, displacement is given by:

\(s = \text{Average Velocity} \times \text{Time}\)

\(s = \frac{u+v}{2} \times t\)

Multiplying both sides by 2 gives:

\(2s = (u+v)t\)

The given option is \(2s = (v - u)t\), which is incorrect because it uses \((v-u)\) instead of \((u+v)\).

Option 3: \(2s = 2ut + at^2\)

Let's look at the standard displacement-time relation:

\(s = ut + \frac{1}{2}at^2\)

Now, let's multiply this entire equation by 2:

\(2 \times s = 2 \times (ut + \frac{1}{2}at^2)\)

\(2s = 2ut + 2 \times \frac{1}{2}at^2\)

\(2s = 2ut + at^2\)

This derived equation exactly matches the given option. Therefore, this is a correct equation of motion, derived directly from one of the standard equations.

Option 4: \(v^2 + u^2 = 2as\)

Let's look at the standard velocity-displacement relation:

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

If we rearrange this equation, we get:

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

The given option is \(v^2 + u^2 = 2as\), which is incorrect. The correct relation involves the difference of the squares of velocities, not the sum.

Conclusion

Based on the analysis, Option 3, \(2s = 2ut + at^2\), is the only equation among the given options that is a correct form derived from the standard equations of uniformly accelerated motion.

Revision Table: Standard Equations of Motion

Additional Information on Kinematics

Kinematics is a branch of classical mechanics that describes the motion of points, bodies (objects), and systems of bodies without reference to the forces that cause them to move. Understanding the equations of motion is crucial for solving problems involving linear motion, projectile motion, and other areas of physics.

These equations are valid only when the acceleration is constant and the motion is in a straight line. For cases where acceleration changes, or motion is not linear, more advanced techniques like calculus are required.

The terms used in the equations represent:

• u: Initial velocity (velocity at the start of the time interval)
• v: Final velocity (velocity at the end of the time interval)
• a: Constant acceleration (rate of change of velocity)
• t: Time interval
• s: Displacement (change in position)