(a)

Given :- Velocity of sand loaded cart $= v$

To Find :- Direction and speed of sand after falling on ground

Solution :- Let the mass of cart be $m _1$

Mass of sand be $m _2$

And, Velocity of sand be $v_2$

Now, By law of conservation of momentum i.e,

$\left( m _1+ m _2\right) v = m _1 \times v + m _2 \times v _2$

$m _1 v + m _2 v = m _1 v + m _2 v _2$

$m _2 v = m _2 v _2$

$\therefore v = v _2$

$\Rightarrow$ Sand is moving in the same direction and with same speed that of cart

$\therefore \quad$ Sand is moving with cart

Hence, Option $A$ (moving with cart) is correct.