Two cars A and B are travelling in the same direction with velocity `upsilon_(1)` and `upsilon_(2) (upsilon_(1)gt upsilon_(2))`. When the car A is at a distance d ahead of the car B, the driver of the car A applied the brake producing a uniform retardation a. There will be no collision when

Options

A. ` d lt ((v_1-v_2)^(2))/(2a) `
B. ` d lt (v_1^(2)-v_2^(2))/(2a) `
C. ` d gt ((v_1-v_2)^(2))/(2a) `
D. ` d gt ( v_1^(2) -v_2^(2)) /( 2a ) `

Answer

C

Explanation

Initial relative velocity ` = v_1-v_2 ,` Final relative velocity =0 Let the cars collde after time t from the instant when begins breaking Then ` ,v_1t-(1)/(2) at^(2) =v_2t+d" "at^(2) +2(v_2+v_1) t+2d =0`
For no collision ,this quadratic it should have no real roots `rArr 4 (v_2 -v_1)^(2) -8ad lt 0`
` rArr d gt ((v_2-v_1)^(2))/(2a) `