Explanation:

To solve this problem, we can consider the forces acting on the lift when it is moving upwards and downwards.

When the lift is moving upwards:
- The tension in the cable provides an upward force, denoted as T.
- The weight of the lift acts downwards, equal to mg (where m is the mass of the lift and g is the acceleration due to gravity).

Therefore, the net force acting on the lift when it is moving upwards is:
Net force = T - mg

When the lift is moving downwards:
- The tension in the cable provides a downward force, denoted as T.
- The weight of the lift still acts downwards, equal to mg.

Therefore, the net force acting on the lift when it is moving downwards is:
Net force = mg - T

Setting up the equation:

Given that the tension in the cable when the lift is moving downwards is half the tension when it is moving upwards, we can write:
T(downwards) = 0.5 * T(upwards)

Substituting this into the net force equation for the downward motion:
mg - T(downwards) = mg - 0.5 * T(upwards)

Simplifying the equation:

Since the mass of the lift (m) and the acceleration due to gravity (g) are common to both sides of the equation, we can cancel them out. This leaves us with:
-T(downwards) = -0.5 * T(upwards)

Multiplying both sides by -1, we get:
T(downwards) = 0.5 * T(upwards)

Relating net force to acceleration:

Using Newton's second law of motion, we know that net force (F_net) is equal to mass (m) multiplied by acceleration (a):
F_net = ma

Since we've already canceled out the mass and acceleration due to gravity, we can rewrite the net force equation for the downward motion as:
ma(downwards) = 0.5 * ma(upwards)

Cancelling out the mass, we get:
a(downwards) = 0.5 * a(upwards)

Conclusion:

The acceleration of the lift when it is moving downwards is half the acceleration when it is moving upwards, which corresponds to option (b) g/3.