Introduction:

The equation S = ut + 1/2at^2 is a formula used to calculate the displacement (S) of an object undergoing constant acceleration. This equation is derived from the basic equations of motion and can be proven using mathematical reasoning.

Proof:

1. Definition of Displacement:
Displacement is the change in position of an object in a particular direction. It can be calculated by subtracting the initial position (u) from the final position (S). Mathematically, S = u + Δx, where Δx represents the change in position.

2. Equation of Motion:
The equation of motion for an object undergoing constant acceleration is given by S = ut + 1/2at^2, where:
- S is the displacement
- u is the initial velocity
- t is the time taken
- a is the constant acceleration

3. Deriving the Equation:
To prove the equation S = ut + 1/2at^2, we start with the definition of displacement:

S = u + Δx

As the object is under constant acceleration, we can express the change in position (Δx) as:

Δx = ut + 1/2at^2

Substituting this into the equation for displacement, we get:

S = u + (ut + 1/2at^2)

Simplifying the expression, we have:

S = ut + 1/2at^2

4. Understanding the Equation:
The equation S = ut + 1/2at^2 can be interpreted as follows:
- The first term, ut, represents the displacement caused by the initial velocity u over time t.
- The second term, 1/2at^2, represents the displacement caused by the constant acceleration a over time t.

Thus, the equation combines the effects of initial velocity and constant acceleration to calculate the total displacement of an object.

Conclusion:

In conclusion, the equation S = ut + 1/2at^2 can be proven by starting with the definition of displacement and applying the concept of constant acceleration. This equation is a fundamental formula used in physics to calculate the displacement of an object undergoing constant acceleration.