A CAR STARTS MOVING RECTILINEARLY FROM REST, FIRST WITH ACCELERATION a=5m/s^2
THEN UNIFORMALLY AND FINALLY DECELERATING AT THE SAME RATE a COMES TO STOP.THE TOTAL TIME OF MOTION EQUALS t=25s.THE AVERAGE VELOCITY AT THAT TIME IS EQUAL TO 72Km/hr. HOW LONG DOES THE CAR MOVE UNIFORMLY.
 

Asked by JAIBHARGAVA200 | 1st Nov, 2014, 02:53: PM

Expert Answer:

begin mathsize 14px style Here space the space car space has space three space motion : Case space 1 : Accelerated space Motion Case space 2 : space Uniform space Motion Case space 3 : space Decelerated space motion  Case space 1 : Let space straight x subscript 1 space end subscript be space the space distance space covered space by space the space car space during space accelerated space motion space in space time space straight t subscript 1 Given space that space acceleration space straight a space equals space 5 straight m divided by straight s squared comma space Since space the space car space starts space from space rest comma space straight u space equals 0 So space from space the space equation space of space motion : straight s space equals ut plus 1 half at squared Here space we space can space write space it space as : straight x subscript 1 space equals 0 plus 1 half cross times 5 cross times straight t subscript 1 squared straight x subscript 1 equals fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus left parenthesis 1 right parenthesis  Case space 2 : Let space straight x subscript 2 space end subscript be space the space distance space covered space by space the space car space during space the space uniform space motion space in space time space straight t subscript 2 Here space the space uniform space velocity space with space with space the space car space travels space is : straight v space equals straight u plus at straight v space equals 0 plus 5 straight t subscript 1 space So space the space distance space covered space by space the space car space during space the space uniform space motion space in space time space straight t subscript 2 space is : straight x subscript 2 equals speed space cross times time space straight x subscript 2 equals 5 straight t subscript 1 straight t subscript 2 space space minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus left parenthesis 2 right parenthesis  Case space 3 : space Let space straight x subscript 3 space end subscript space be space the space distance space covered space by space the space car space during space decelerated space motion space in space time space straight t subscript 3 Here space the space initial space velocity space of space the space car space equals space 5 straight t subscript 1 Given space retardation comma space straight a space equals minus 5 space straight m divided by straight s squared So space we space can space write space the space equation space of space motion space as : straight x subscript 3 space equals 5 straight t subscript 1 straight t subscript 3 minus 1 half cross times 5 cross times straight t subscript 3 squared straight x subscript 3 equals 5 straight t subscript 1 straight t subscript 3 minus fraction numerator 5 straight t subscript 3 squared over denominator 2 end fraction space minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus left parenthesis 3 right parenthesis end style
begin mathsize 14px style Also space given space that space the space average space velocity space of space the space motion space equals space 72 space km divided by hr space equals 72 cross times 5 over 18 equals 20 space straight m divided by straight s Total space time space equals space 25 space straight s Hence space we space can space write space that space average space velocity equals fraction numerator Total space distance space over denominator Toatl space time end fraction 20 space straight m divided by straight s space equals fraction numerator straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 over denominator straight t subscript 1 plus straight t subscript 2 plus straight t subscript 3 end fraction minus minus minus minus minus minus minus minus minus minus minus left parenthesis 4 right parenthesis  straight x subscript 1 plus straight x subscript 2 plus straight x subscript 3 space equals fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction plus 5 straight t subscript 1 straight t subscript 2 plus 5 straight t subscript 1 straight t subscript 3 minus fraction numerator 5 straight t subscript 3 squared over denominator 2 end fraction  Further space for space the space decelerated space motion space we space can space wite space that space straight v space equals straight u plus at here space straight v space equals straight o space rightwards double arrow space 5 straight t subscript 1 equals 5 straight t subscript 3 rightwards double arrow space straight t subscript 1 equals straight t subscript 3 Given space that space straight t subscript 1 plus straight t subscript 2 plus straight t subscript 3 equals 25 rightwards double arrow space 2 straight t subscript 1 plus straight t subscript 2 equals 25 straight t subscript 2 equals 25 minus 2 straight t subscript 1 space minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus minus left parenthesis 5 right parenthesis Substituting space all space values space in space left parenthesis 4 right parenthesis space we space get : 20 space equals space fraction numerator open parentheses fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction plus 5 straight t subscript 1 straight t subscript 2 plus 5 straight t subscript 1 straight t subscript 3 minus fraction numerator 5 straight t subscript 3 squared over denominator 2 end fraction close parentheses over denominator 25 end fraction 500 equals fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction plus 5 straight t subscript 1 straight t subscript 2 plus 5 straight t subscript 1 straight t subscript 3 minus fraction numerator 5 straight t subscript 3 squared over denominator 2 end fraction As space straight t subscript 1 equals straight t subscript 3 : 500 space equals space fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction plus 5 straight t subscript 1 straight t subscript 2 plus 5 straight t subscript 1 squared minus fraction numerator 5 straight t subscript 1 squared over denominator 2 end fraction Therefore space it space becomes : 500 space equals space 5 straight t subscript 1 straight t subscript 2 plus 5 straight t subscript 1 squared 500 space equals space 5 straight t subscript 1 left parenthesis 25 minus 2 straight t subscript 1 right parenthesis space plus 5 straight t subscript 1 squared 100 space equals space 25 space straight t subscript 1 minus 2 straight t subscript 1 squared plus straight t subscript 1 squared straight t subscript 1 squared minus 25 space straight t subscript 1 plus 100 space equals space 0 On space solving space this space quadratic space equation space we space get : straight t subscript 1 equals 20 space sec space and space straight t subscript 1 equals 5 space sec As space straight t subscript 1 space cant space be space 2 space 0 sec space as space it space does space not space satisfy space the space conditions space that space straight t subscript 1 plus straight t subscript 2 plus straight t subscript 3 equals 25 space space and space space straight t subscript 1 equals straight t subscript 3 space we space have space straight t subscript 1 space equals space 5 sec So space the space time space of space uniform space motion space straight t subscript 2 space equals 25 space minus space 2 straight t subscript 1 straight t subscript 2 space equals 25 space minus 2 cross times 5 straight t subscript 2 equals 25 minus 10 space space space equals 15 space sec end style
Hence for 15 sec teh car moves uniformly.

Answered by Jyothi Nair | 2nd Nov, 2014, 11:53: AM