When a ball is dropped into a lake from a height $$4.9$$ m above the water level, it hits the water with a velocity $$v$$ and then sinks to the bottom with the constant velocity $$v$$. It reaches the bottom of the lake $$4.0$$ s after it is dropped. The approximate depth of the lake is

A ball is dropped from a height of $$4.9$$ m above the water level. It hits the water with velocity $$v$$ and then sinks with the same constant velocity $$v$$. The total time from drop to reaching the bottom is $$4.0$$ s.

Find the velocity when the ball hits the water.

Using $$v^2 = u^2 + 2gh$$ with $$u = 0$$, $$g = 9.8$$ m/s², $$h = 4.9$$ m:

$$v^2 = 2 \times 9.8 \times 4.9 = 96.04$$

$$v = 9.8$$ m/s

Find the time taken to fall through air.

Using $$v = u + gt$$:

$$9.8 = 0 + 9.8 \times t_1$$

$$t_1 = 1$$ s

Find the time spent sinking in water.

$$t_2 = 4.0 - 1.0 = 3.0$$ s

Find the depth of the lake.

The ball sinks with constant velocity $$v = 9.8$$ m/s:

$$d = v \times t_2 = 9.8 \times 3.0 = 29.4$$ m

The approximate depth of the lake is $$29.4$$ m.

The correct answer is Option D.