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


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


Derive the equation of motion which is independent of t.


Derive the third equation of motion?


Prove that   $2aS = v_f^2 - v_i^2$

Difficulty: Hard

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 = ${vav}\times t$

S = $\frac{vf \: + \: vi}{2}\times t$ ……..(1)

From the first equation of motion find the value of t.      

vf = vi +at      Or     t = $S=\frac{vf \: - \: vi}{a}$

Putting the value of Vf in equation (1).

S = $\frac{vf \: + \: vi}{2}\times\frac{vf \: - \: vi}{a}$

2as = $(vf+vi)\times (vf - vi)$   by using formula   2as = $(a+b)(a-b)=a^{2}-b^{2}$

2as = $vf^{2}-vi^{2}$


Second Method (Graphical method)

Third equation of motion:

In the speed-time graph shown in the figure, the total distance S traveled by the body is given by the total area OABD under the graph.


Total area $OABD = S=\left(\frac{OA + BD}{2} \right)\times OD$

Or          $2S = (OA + BD) x OD$


Multiply both sides by BC/OD, we get:               


$\frac{BC}{OD} =a$

$2S\times\frac{BC}{OD} =\left(OA + BD\right)\times OD\times\frac{BC}{OD}$

$2S\times\frac{BC}{OD} =\left(OA + BD\right)\times BC$ .........(1)




Putting the value in the above equation (1), we get

$2S\times a =\left(Vi + Vf\right)\times \left(Vf - Vi\right)$

$2aS = Vf^{2} + Vi^{2}$





  • To convert ms-1 to kmh-1

1 ms-1 = 0.001km x 3600 = 3.6 kmh-1

Thus, multiply speed in ms-1 by 3.6 to get speed in km-1 e.g.,

20 ms-1= 20 x 3.6 kmh-1=72 kmh-1


  • To convert kmh-1 to ms-1

$1 kmh^{-1}= \frac{1000m}{60x60s} =\frac{10}{36}ms^{-1}$

Thus, multiply speed in kmh-1 by  to get speed in ms-1 e.g.

$50 kmh^{-1}= 50 \times \frac{10}{36} ms^{-1}= 13.88 ms^{-1}$


  • To convert ms-2 to kmh-2

Multiply acceleration in ms-2 by $\frac{3600\times3600}{1000} = 12960$ to get its value in kmh-2


  • To convert km-2 to ms-2

Divide acceleration in kmh-2 by 12960 to get its value in ms-2



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