Question
Establish the kinematic equation v2 - u2 = 2as from velocity-time graph for a uniformly accelerated motion.
✓
Answer
The velocity-time graph for uniformly accelerated motion has been shown in Fig. with initial velocity at t = 0 as u and final velocity at time t as v. Then area under the v-t graph gives the value of total displacement in the given time.
Hence, displacement of moving particle in time t = area of trapezium $\Delta \text{ABC}$
$\text{s}=\frac{1}{2}(\text{OA}+\text{CB})\times\text{OC}$
$=\frac{1}{2}(\text{u}+\text{v})\times\text{t}$
However, from definition of acceleratiory we know that
$\text{a}=\frac{\text{v}-\text{u}}{\text{t}}$ or $\text{t}=\frac{\text{v}-\text{u}}{\text{a}}$
Substituting this value of time t in equation (i), we get
Displacemen $\text{s}=\frac{1}{2}(\text{u}+\text{v})\times\frac{\text{v}-\text{u}}{\text{a}}$
$\frac{(\text{v}^2-\text{u}^2)}{2\text{a}}$
$\Rightarrow 2\text{as}=\text{v}^2-\text{u}^2$
$\text{v}^2=\text{u}^2+2\text{as}$
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