A particle of mass m moving with horizontal speed 6 m/sec. If m<

The correct option is 2 m/sec in original direction.

To find the speed of the lighter particle after a one-dimensional elastic collision, we use the standard velocity equation for elastic collisions. Let the lighter particle have mass $m$ and initial velocity $u_1=6\text{m/s}$, and the heavier particle have mass $M$ and initial velocity $u_2$ in the same direction.

The final velocity $v_1$ of the lighter particle after the elastic collision is given by:
$v_1=\left(\frac{m-M}{m+M}\right)u_1+\left(\frac{2M}{m+M}\right)u_2$

Given that the mass of the lighter particle is much smaller than the heavier particle ($m\ll M$), we can approximate the mass ratios by taking the limit as $\frac{m}{M}\to 0$:
$\frac{m-M}{m+M}\approx -1$
and
$\frac{2M}{m+M}\approx 2$

Substituting these approximations into the velocity equation gives:
$v_1\approx -u_1+2u_2$

Considering the scenario where the heavier mass $M$ moves with a velocity of $u_2=4\text{m/s}$ in the same direction, we substitute the values:
$v_1=-6+2\left(4\right)$
$v_1=-6+8=2\text{m/s}$

Since the resulting velocity is positive, the lighter particle moves with a speed of 2 m/sec in original direction after the collision.