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

                    OR

Which equation of motion establishes the relationship between S, a, Vi and Vf?

                    OR

Derive the equation of motion which is independent of t.

                   OR

Derive the second equation of motion?

                   OR

Prove that   S = v1 t + $\cfrac{1}{2}$ at2

Difficulty: Medium

Suppose a body is moving with initial velocity vi and after a certain time t its velocity becomes vf 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.                 Vf = Vi + at   

Putting the value of Vf 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 at2

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

                                           = Vi 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 = Vit + ½ t x at

                          S = Vit + ½ at2

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