Two paricles A and B start simultaneously from the same point and move in a horizontal plane. A has an initial velocity `u_(1)` due east and acceleration `a_(1)` due north. B has an intial velocity `u_(2)` due north and acceleration `a_(2)` due east.

Options

A. Their paths must intersect at some point
B. They must collide at some point
C. They will collide only if `a_1u_1=a_2u_2`
D. If `u_1 gt u_2` and `a_1 lt a_2`, the particles will have the same speed at some point of time

Answer

A, C, D

Explanation

Let Co-ordinate of A and B are `(x_1, y_1) and (x_2, y_2)` respectively at time t \`rArr x_(1)=u_(1)t, y_(1)=1/2 a_(1)t^(2) & x_(2)=1.2 a_(2)t^(2), y_(2)=u_(2)t`
(As acceleration of A is along y-axis, its path is parabola opening towards positive y-axis and of B is parabola opening towards positive x-axis their path will intersect.)
For collision, same position at same time `rArr x_(1)=x_(2) and y_(1)=y_(2)` at same t
`rArr u_(1)t=1/2 a_(2)t^(2) and 1/2 u_(1)t^(2)=u_(2)t rArr t=(2u_(1))/(a_(2)) and t=(2u_(2))/(a_(1))`
`rAr (2u_(1))/(a_(2))=(2u_(2))/(a_(1)) rArr a_(1)u_(1)=a_(2)u_(2)l v_(x_(1))=u_(1),v_(y_(1))=a_(1)t rArr v_(1)=sqrt(u_(1)^(2)+a_(1)^(2)t^(2)).......ie,` speed of at time t.
Similarly speed of B at time t is `sqrt(u_(2)^(2)+a_(2)^(2)t^(2))`, Speed are equal `rArr v_(1)^(2)-v_(2)^(2)=0 rArr u_(1)^(2)-u_(2)^(2)=(a_(2)^(2)-a_(1)^(2))t^(2)` both sides are positive if `u_(1) gt u_(2) and a_(1) lt a_(2)`