Derivation of the Second Equation of Motion

The second equation of motion relates the final velocity of an object to its initial velocity, acceleration, and displacement. It can be derived using the kinematic equations and basic principles of motion.

Key Points:
- The second equation of motion is given by: v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the displacement.
- This equation is applicable when the object undergoes uniform acceleration.

Derivation:

Step 1: Define the variables
- Let u be the initial velocity of the object.
- Let v be the final velocity of the object.
- Let a be the constant acceleration experienced by the object.
- Let s be the displacement of the object.

Step 2: Apply the first equation of motion
The first equation of motion states: v = u + at.
- Rearrange this equation to solve for t: t = (v - u) / a.

Step 3: Substitute the expression for t into the equation for displacement
The equation for displacement is given by: s = ut + (1/2)at^2.
- Substitute the expression for t from Step 2 into the equation for displacement: s = u((v - u) / a) + (1/2)a((v - u) / a)^2.
- Simplify: s = (uv - u^2) / a + (v - u)^2 / (2a).
- Multiply through by a to eliminate the denominator: as = uv - u^2 + (v - u)^2 / 2.

Step 4: Rearrange the equation
- Expand the square term: as = uv - u^2 + (v^2 - 2uv + u^2) / 2.
- Simplify: 2as = 2uv - 2u^2 + v^2 - 2uv + u^2.
- Combine like terms: 2as = v^2 - u^2.
- Rearrange the equation: v^2 = u^2 + 2as.

Conclusion:
The second equation of motion, v^2 = u^2 + 2as, is derived by applying the kinematic equations and basic principles of motion. It relates the final velocity of an object to its initial velocity, acceleration, and displacement. This equation is useful in various physics problems involving uniformly accelerated motions.