question_answer

A particle experiences constant acceleration for 20 seconds after starting from rest. If it travels a distance \[{{\text{S}}_{\text{1}}}\]in the first 10 seconds and distance \[{{\text{S}}_{\text{2}}}\] in the next 10 seconds, find the relation between \[{{\text{S}}_{\text{1}}}\]and\[{{\text{S}}_{\text{2}}}\].

Answer:

                Let at an instant t,                 \[\upsilon\] be the velocity of the moving particle and s be the distance travelled by the particle. As per question. \[\text{s }=\text{ }\upsilon \text{t}/\text{2}\]                                                                                                    ...(i) Differentiating it w.r.t. time t, we have                                 \[\frac{ds}{dt}=\frac{1}{2}\frac{d\upsilon }{dt}\times t+\frac{\upsilon }{2}\] Or                           \[\upsilon =\frac{1}{2}a\times +\frac{\upsilon }{2}\] or at = \[\upsilon \] Differentiating it again w.r.t. time t, we have \[\frac{da}{dt}\times \text{t }+\text{ a }=\frac{d\upsilon }{dt}=\text{ a}\]or\[\frac{da}{dt}\times \text{t =}\,\text{0}\]or\[\frac{da}{dt}\text{=}\,\text{0}\] Therefore ; a = a constant.