question_answer

A point moves with a uniform acceleration and denote the average velocities in the three successive intervals of time \[{{t}_{1}}\], \[{{t}_{2}}\] and\[{{t}_{3}}\] . Find the ration of\[\left( {{\upsilon }_{1}}-{{\upsilon }_{2}} \right)\] and \[\left( {{\upsilon }_{2}}-{{\upsilon }_{3}} \right)\].

Answer:

Let the point starts moving from O with a uniform acceleration a along a st. line. It reaches at locations A, B and C at timings and respectively Fig. 2(HT).6. Let \[\upsilon '\],\[\upsilon ''\], \[\upsilon '''\]be the velocity of point at A, B and C respectively \[\therefore \]   Initial velocity of point at O, u = 0 Using the formula; \[\upsilon =u+at\] Taking motion from O to A, we have ; \[\upsilon '=0+a{{t}_{1}}=a{{t}_{1}}\] Taking motion from O to B, we have; \[\upsilon ''=0+a({{t}_{1}}+{{t}_{2}})=a({{t}_{1}}+{{t}_{2}})\] Taking motion form O to C, we have \[\upsilon '''=0+a({{t}_{1}}+{{t}_{2}}+{{t}_{3}})=a({{t}_{1}}+{{t}_{2}}+{{t}_{3}})\] \[\therefore \] Average velocity in interval of time \[{{t}_{1}}\]. \[{{\upsilon }_{1}}=\frac{0+\upsilon '}{2}=\frac{a{{t}_{1}}}{2}\]                                              ?? (1) Average velocity in interval of time \[{{t}_{2}}\], \[{{\upsilon }_{2}}=\frac{\upsilon '+\upsilon ''}{2}=\frac{a{{t}_{1}}+a\left( {{t}_{1}}+{{t}_{2}} \right)}{2}=a{{t}_{1}}+\frac{1}{2}a{{t}_{2}}\]  ??.. (2) Average velocity in interval of time \[{{t}_{3}}\], \[{{\upsilon }_{3}}=\frac{\upsilon ''+\upsilon '''}{2}=\frac{a\left( {{t}_{1}}+{{t}_{2}} \right)+a\left( {{t}_{1}}+{{t}_{2}}+{{t}_{3}} \right)}{2}=a\left( {{t}_{1}}+{{t}_{2}} \right)+\frac{1}{2}a{{t}_{3}}\]                    ??. (3) \[\therefore \]   \[{{\upsilon }_{2}}-{{\upsilon }_{1}}=\left( a{{t}_{1}}+\frac{1}{2}a{{t}_{2}} \right)-\frac{a{{t}_{1}}}{2}=\frac{a}{2}\left( {{t}_{1}}+{{t}_{2}} \right)\]                                        ??.. (4) and \[{{\upsilon }_{3}}-{{\upsilon }_{2}}=\left( a({{t}_{1}}+{{t}_{2}})+\frac{1}{2}a{{t}_{3}} \right)-\left[ a{{t}_{1}}+\frac{1}{2}a{{t}_{2}} \right]=\frac{a}{2}\left( {{t}_{2}}+{{t}_{3}} \right)\]                     ??. (5) From (4) and (5), \[\frac{{{\upsilon }_{2}}-{{\upsilon }_{1}}}{{{\upsilon }_{3}}-{{\upsilon }_{2}}}=\frac{\left( {{t}_{1}}+{{t}_{2}} \right)}{\left( {{t}_{2}}+{{t}_{3}} \right)}\]or \[\frac{{{\upsilon }_{1}}-{{\upsilon }_{2}}}{{{\upsilon }_{2}}-{{\upsilon }_{3}}}=\frac{{{t}_{1}}+{{t}_{2}}}{{{t}_{2}}+{{t}_{3}}}\]