1. Collinearity Condition

Points A, B, and C are separated by equal horizontal distances $d$. For them to remain collinear at all times, the height of B must be the average of the heights of A and C.

$$ y_B(t) = \frac{y_A(t) + y_C(t)}{2} $$

2. Kinematics of A and C

Point A: Moves vertically upwards with constant velocity $u$.

$$ y_A(t) = ut $$

Point C: Moves vertically downwards with constant acceleration $a$ (starting from rest).

$$ y_C(t) = -\frac{1}{2}at^2 $$

3. Determining Motion of B

Substituting $y_A$ and $y_C$ into the collinearity condition:

$$ y_B(t) = \frac{1}{2} \left( ut – \frac{1}{2}at^2 \right) = \frac{u}{2}t – \frac{a}{4}t^2 $$

Velocity of B: Differentiating position with respect to time.

$$ v_B(t) = \frac{dy_B}{dt} = \frac{u}{2} – \frac{a}{2}t $$

At $t=0$, initial velocity is $u/2$ upwards.

Acceleration of B: Differentiating velocity with respect to time.

$$ a_B = \frac{dv_B}{dt} = -\frac{a}{2} $$

The acceleration is constant $a/2$ downwards.