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Show that when a particle with uniform acceleration, the distances described in consecutive equal intervals of time are in A.P. |
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Answer» Solution :Let us CONSIDER equal intervals of time,each having value t.Suppose u be the VELOCITY at the beginnig of the first time interval and a the uniform acceleration. DISTANCE travelled during the first n intervals [i.e., In time (n-1)t] `S_(1)=u(nt)+(1)/(2)(nt)^(2)a` Distance travelled during the first (n-1) interval [i.e., in time (n-1)t] `S_(2)=u(n-1)t+(1)/(2)a(n-1)^(2)t^(2)` Distance travelled in the nthtime interval=`S_(1)-S_(2)` `=(unt+(1)/(2)an^(2)t^(2))-[unt-ut+(1)/(2)an^(2)t^(2)-ant^(2)+(1)/(2)at^(2)]` `=ut+ant^(2)-(1)/(2)at^(2)=ut+(1)/(2)(2n-1)at^(2)` PUTTING n=1,2,3................, we get DISTANCES travelled in 1st,2nd,3rd.........we get distances travelled in 1st,2nd,3rd....... intervals to be `ut+(1)/(2)at^(2),ut+(5)/(2)at^(2)`......... Thus, distance described in consecutive equal intervals of time are in A.P. with common difference `at^(2)` |
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