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Home / Doubts and Solutions/NEET/Physics

a particle starts from rest and travels a distance s with uniform acceleration,then it travels a distance 2s with uniform speed,finally it travels a distance 3s with uniform retardation and comes to rest.if the complete motion of the particle is a straight line then the ratio of its average velocity to maximum velocity is

Asked by 443879616104558 21st August 2018, 10:25 AM
Answered by Expert
Answer:
Velocity after travelling a distance with acceleration a is obtained from equation of motion " v2 = u2 +2aS ",
where u is initial velocity which is zero in this case because particle starts from rest.
v is final velocity, a is acceleration and S is the distance travelled.
 
Hence velocity after travelling a distance S :-  begin mathsize 12px style v space equals space square root of 2 space a space S end root end style......................(1)
time taken to travel the distance S starting from rest :-  S = (1/2)at2 or begin mathsize 12px style t subscript 1 space equals space square root of fraction numerator 2 S over denominator a end fraction end root end style................(2)
If particle travels distance 2S with uniform speed given by eqn.(1), then time to complete the travel: - t2 begin mathsize 12px style equals space fraction numerator 2 S over denominator square root of 2 a S end root end fraction space equals space square root of fraction numerator 2 S over denominator a end fraction end root end style................(3)
When the particle travels with the initial velocity as given by eqn.(1) and subjected to a retardation for a
distance 3S and stooped,then the retardation is calculated as follows :-
 
begin mathsize 12px style 0 space equals space 2 space a space S space minus space 2 cross times a subscript r cross times 3 S space space o r space a subscript r space equals space a over 3 end style
time to reach zero velocity starting from intial velocity given by eqn.(1) is :-  t3 = begin mathsize 12px style fraction numerator square root of 2 a S end root over denominator begin display style bevelled a over 3 end style end fraction space equals space 3 space square root of fraction numerator 2 S over denominator a end fraction end root end style.......................(4)
total distance travelled = S+2S+3S = 6S
 
total time = t1 + t2 + t3 = begin mathsize 12px style 5 square root of fraction numerator 2 S over denominator a end fraction end root end style..............................(5)
average velocity = begin mathsize 12px style fraction numerator t o t a l space d i s tan c e over denominator t o t a l space t i m e end fraction space equals space fraction numerator 6 S over denominator open parentheses 5 cross times square root of begin display style bevelled fraction numerator 2 S over denominator a end fraction end style end root close parentheses end fraction equals fraction numerator 6 over denominator 5 square root of 2 end fraction square root of a S end root space equals 3 over 5 square root of 2 a S end root end style...................(6)
maximum velocity is given by eqn.(1)
ratio between average velocity to maximum velocity is obtained from eqn.(6) and eqn.(1). This ratio is 3/5
Answered by Expert 21st August 2018, 11:58 AM
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