A train starts from a station P with a uniform acceleration a1, for some distance and then goes with uniform retardation a2 for some more distance to come to rest at the station Q. The distance between the stations P and Q is 4Km and the train takes 4 minutes to complete this journey, then 1/a1plus1/a2=

Asked by Petulla Mishra | 4th Nov, 2015, 10:10: AM

Expert Answer:

There are two case in given question:
1. When train moves with uniform acceleration
2. When train moves with uniform retardation.
 
For the 1st case,
Let the train start from a station P with uniform accleration a1, with initial velocity u1 and t1 be the time when it reaches at some distance s1
From kinematic eqution of motion we have,
v1=u1+a1t1         (u1=0 Initialy at rest)
v1=a1t1     .... (i)
begin mathsize 14px style straight a subscript 1 equals straight v subscript 1 over straight t subscript 1 space space space space... space left parenthesis i i right parenthesis end style
For the 2nd case the final velocity of first interval will be intial velocity of second.
Let a2 be uniform retardation, t2 be the time when it travels a distance s2 and comes to rest .
From kinematic equation of motion we have,
v= u- a2t= v- a2t2    (v= 0 finaly comes at rest)
v= a2t2  ... (iii) 
 
begin mathsize 14px style straight a subscript 2 equals straight v subscript 1 over straight t subscript 2 space space space space... space left parenthesis iv right parenthesis end style
Using kinematic equation of motion,
begin mathsize 14px style straight s subscript 1 equals straight u subscript 1 straight t subscript 1 plus 1 half a subscript 1 straight t subscript 1 superscript 2 straight s subscript 1 equals 1 half straight a subscript 1 straight t subscript 1 superscript 2 equals 1 half straight v subscript 1 over straight t subscript 1 cross times straight t subscript 1 superscript 2 straight s subscript 1 equals 1 half straight v subscript 1 straight t subscript 1 space space space... space left parenthesis straight v right parenthesis Also comma straight s subscript 2 equals straight u subscript 2 straight t subscript 2 minus 1 half straight a subscript 2 straight t subscript 2 superscript 2 From space left parenthesis straight i right parenthesis comma space we space get comma straight s subscript 2 equals straight v subscript 1 straight t subscript 2 minus 1 half straight v subscript 1 over straight t subscript 2 cross times straight t subscript 2 superscript 2 straight s subscript 2 equals 1 half straight v subscript 1 straight t subscript 2 straight s equals straight s subscript 1 plus straight s subscript 2 equals 1 half straight v subscript 1 straight t subscript 1 plus 1 half straight v subscript 1 straight t subscript 2 straight s equals 1 half straight v cross times open parentheses straight t subscript 1 plus straight t subscript 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket because straight t subscript 1 equals straight t subscript 2 right square bracket space 4 cross times 1000 equals 1 half cross times straight v cross times 4 cross times 60 space space space space space space space space space space space space space space space space space left square bracket because straight t subscript 1 plus straight t subscript 1 equals 4 space min equals 4 cross times 60 equals 240 space and space straight s equals 4 km equals 4 cross times 1000 equals 4000 straight m right square bracket straight v equals 8000 over 240 equals 33.33 space straight m divided by straight s From space eqn space left parenthesis straight i right parenthesis space and space left parenthesis iv right parenthesis comma space we space get comma 1 over straight a subscript 1 plus 1 over straight a subscript 2 equals straight t subscript 1 over straight v subscript 1 plus straight t subscript 2 over straight v subscript 1 equals fraction numerator straight t subscript 1 plus straight t subscript 2 over denominator straight v subscript 1 end fraction equals fraction numerator 240 over denominator 33.33 end fraction equals 7.2 therefore 1 over straight a subscript 1 plus 1 over straight a subscript 2 equals 7.2 end style

Answered by Faiza Lambe | 4th Nov, 2015, 11:06: AM