The third equation of motion relates the final velocity (v), initial velocity (u), acceleration (a), and displacement (s) of an object in uniform motion. Here's how to derive it:

1. Start with the first equation of motion:

* v = u + at

This equation tells us the final velocity (v) is equal to the initial velocity (u) plus the product of acceleration (a) and time (t).

2. Use the second equation of motion:

* s = ut + (1/2)at²

This equation tells us the displacement (s) is equal to the initial velocity (u) multiplied by time (t) plus half the product of acceleration (a) and the square of time (t²).

3. Express time (t) in terms of other variables:

* From the first equation, we can rearrange to solve for time:

* t = (v - u) / a

4. Substitute the expression for time (t) into the second equation:

* s = u[(v - u) / a] + (1/2)a[(v - u) / a]²

5. Simplify the equation:

* s = (uv - u²) / a + (1/2)a[(v² - 2uv + u²) / a²]

* s = (uv - u²) / a + (v² - 2uv + u²) / (2a)

* s = (2uv - 2u² + v² - 2uv + u²) / (2a)

* s = (v² - u²) / (2a)

6. Rearrange the equation to get the third equation of motion:

* v² = u² + 2as

Therefore, the third equation of motion is v² = u² + 2as.

This equation is useful for calculating the final velocity of an object if you know its initial velocity, acceleration, and displacement. It can also be used to calculate the displacement of an object if you know its initial and final velocity and acceleration.