**Derive the second equation of motion for uniformly accelerated rectilinear motion.**

** OR**

**Which equation of motion establishes the relationship between S, a, V _{i }and V_{f}?**

** OR**

**Derive the equation of motion which is independent of t.**

** OR**

**Derive the second equation of motion?**

** OR**

**Prove that S = v _{1} t + $\cfrac{1}{2}$ at^{2}**

Difficulty: Medium

**Suppose a body is moving with initial velocity v _{i} and after a certain time t its velocity becomes v_{f} then the total distance S covered in time t is given by**

$S = v_{av} \times t$

\begin{equation} S = \dfrac{vf + Vi}{2} \times t \end{equation}

From the first equation of motion. V_{f} = V_{i }+ at

Putting the value of V_{f} in equation (1).

$\dfrac{vf + at vi}{2} \times t$

$\dfrac{2vi+ at}{2} \times t$

$\dfrac{2vit+ at2}{2} \times t$

$\dfrac{2vit}{2}$ + $\dfrac{at2}{2}$

S = V.t + ½ at^{2}

**Second Method (Graphical method):**

**The second equation of motion:**

In the speed-time graph shown in the figure, the total distance S travelled by the body is equal to the total area OABD under the graph. That is

Total distance S = area of (rectangle OACD + triangle ABC)

Area of rectangle OACD = OA x OD

= V_{i} x t

Area of triangle ABC = ½ (AC x BC)

= ½ t x at

Since Total area OABD = area of rectangle OACD + area of triangle ABC

Putting values in the above equation, we get

S = V_{i}t + ½ t x at

S = V_{i}t + ½ at^{2}

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