Question
Establish the kinematic equation $\text{s}=\text{ut}+\frac{1}{2}\text{at}^2$ from velocity-time graph for a uniformly accelerated motion.
✓
Answer
Let AB be a velocity-time graph for uniformly accelerated motion with initial velocity u at time t = 0 and acceleration of the particle under motion being given by $\text{a}=\tan\theta =\frac{\text{BD}}{\text{AD}}$ We know that area under the v-t graph gives the value of displacement during that time. $\therefore$ Displacement of particle in time t will be s = area under v-t graph = area OABC = Area of rectangle OADC + area of triangle ADB $=\text{OA}\times\text{OC}+\frac{1}{2}\text{AD}\times\text{DB}\\=\text{u}\times\text{t}+\frac{1}{2}(\text{AD})\times\Big(\frac{\text{AD}\times\text{DB}}{\text{AD}}\Big)$ $=\text{ut}+\frac{1}{2}(\text{AD})^2\times\Big(\frac{\text{DB}}{\text{AD}}\Big)\text{s}=\text{ut}+\frac{1}{2}\text{t}^2\text{a}$ $\therefore \text{s}=\text{ut}+\frac{1}{2}\text{at}^2$
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