2 cars A and B are travelling in the same direction with velocities V_A and V_B (V_A > V_B), when the car A is at a distance s behind the car B, the driver of the car A applies brakes producing a uniform retardation a and there will be no collision when

a) s<(V_A-V_B)²/2a                                                                                             b) s<(V_A-V_B)²/2a only

c)s>(V_A-V_B)²/2a                                                                                               d)s<=(V_A-V_B)²/2a  

Guess work will still fetch you 2 points and correctly explained answer will bring you 5 points. So, keep guessing and fetching points.

where is it?

option c.we consider car B to be at rest and car A moving with a velocity v_A-v_B.

distance covered before stopping is (v_A-v_B)²/2a.

so, to prevent it from hitting car B,s>(v_A-v_B)²/2a.

is it (a)

hey friends i think that none of options given is correct.according to me collision will not take place if driver of car A anyhow reduces his speed to Vb within the distance s so s>= Vb^2 - Va^2/ 2a

Mr. S Darwin you made a minute mistake while solving the 3rd step. If you solve it correctly and you will find s = RHS of Darwin's fourth step.Then if s = RHS then collision is sure to take place. Therefore it has to be greater then it. And Charanjit a very similar question was there in JEE and the same 4 options were there as well. And they don't make such silly mistakes(of giving incorrect option). For following correct procedure DarWin is rated with 2 points.

 darvin has taken vb=0 but there is nowhere in q that b was stopped. thats y ans is comin wrong !! 

 by d way i think c is correct !!

c) correct.

Explaination

va - vb relative velocity.

d(va - vb)/dt = a

va - vb is the relative velocity or it can be treated as the speed with which  s is reduced.

so

So this relative velocity has to become 0 before the car behind runs out of s. The distance in which va-vb becomes 0 lets call it s'

if s' < s collition can  be avoided

hence (va-vb)^2/2a = s'

hence 3) is correct

RATES PLEASE

 Damn goiit !! Didnt post my edit

 if u cant find the solution, change the question !!

the rel.velocity between da cars=(Va-Vb)

and as per the formula=v^2-u^2=2as

so, (Va-Vb)^2=2as but

for no collision

(Va-Vb)^2/2a<s ans option 3 is correct

Nikita, Darwin took VB = 0 because you can't proceed without stopping B. Piyush who said that i cant find the answer. if i can correct some of the steps mentioned by others that means i already know the answer.If you can't solve it then reply. I sahl give you all the steps. You can go through the replies of Missionadmission or myself. If its still not clear than you are free to ask back.