Two motor cars on the same line approach each other with velocities u1 and u2 respectively. When each is seen from the other, the distance between them is x. If f1 and f2 to be the maximum retardation of the two cars then a collision can be just avoided then at which condition collision can just be avoided?

Two motor cars on the same line approach each other with velocities u1 and u2 respectively. When each is seen from the other, the distance between them is x. If f1 and f2 to be the maximum retardation of the two cars then a collision can be just avoided then at which condition collision can just be avoided? Correct Answer (u12f2 + u22f1) = 2f1f2(x)

By question the two motor cars approach each other along the same line. Let P and Q be the positions of the cars on the line when each is seen from the other, where PQ = x. It is evident that a collision can be just avoided, if the two cars stop at the point somewhere between P and Q that is if velocities of both the motor cars at O are zero. Since the initial velocity of the motor car is at P is u1 and it comes to rest at O, hence, its equation is 0 = u12 – 2f1(PO) Or (PO) = u12/2f1 Again, the initial velocity of the motor car at Q is due to and it comes to rest at O; hence its equation of motion is, 0 = u22 – 2f2(QO) Or (QO) = u22/2f2 Now, x = PQ = PO + OQ = u12/2f1 + u22/2f2 Or u12f2 + u22f1)/2f1f2 = x Thus, (u12f2 + u22f1) = 2f1f2(x).