The boxes of masses 2 kg and 8 kg are connected by a massless string passing over smooth pulleys. Calculate the time taken by box of mass 8 kg to strike the ground starting from rest. (g = 10 m s$$^{-2}$$)

Let the tension in the string connected to the $$2\text{ kg}$$ mass be $$T$$.

$$\text{Tension on the movable pulley holding the } 8\text{ kg} \text{ mass} = 2T$$

$$\text{Constraint relation: } 2 \cdot a_1 = a_2 \implies a_{2\text{kg}} = 2a,\ a_{8\text{kg}} = a$$

Equations of motion:

$$8g - 2T = 8a \implies 4g - T = 4a$$

$$T - 2g = 2(2a) \implies T - 2g = 4a$$

Adding the equations:  $$2g = 8a \implies a = \frac{g}{4} = \frac{10}{4} = 2.5\text{ m/s}^2$$

Time taken to strike the ground ($$h = 20\text{ cm} = 0.2\text{ m}$$):

$$h = \frac{1}{2}at^2 \implies 0.2 = \frac{1}{2}(2.5)t^2$$

$$t^2 = \frac{0.4}{2.5} = 0.16 \implies t = 0.4\text{ s}$$