A rock is dropped from a tree of height 22 m into a lake (depth 4.0 m) below. After entering the water, the rock floats gently down through the water at a constant speed of 1.4 m/s to the bottom of the lake. What is the total elapsed time?

h = 0.5g*t1^2 = 22 m.

g = 9.8 m/s^2.
t1 = ?.

1.4m/s * t2 = 4 m.
t2 = ?.

T = t1 + t2.

To calculate the total elapsed time, we need to find the time it takes for the rock to fall from the tree to the surface of the water, as well as the time it takes to sink to the bottom of the lake.

Firstly, let's calculate the time it takes for the rock to fall from the tree to the water's surface.

We can use the equation for free fall:

h = (1/2) * g * t^2

Where h is the height, g is the acceleration due to gravity (9.8 m/s^2), and t is the time.

Rearranging the equation, we can solve for t:

t = sqrt((2 * h) / g)

Substituting the given values:

t = sqrt((2 * 22) / 9.8)
t ≈ 2.01 seconds

So, it takes approximately 2.01 seconds for the rock to fall from the tree to the water's surface.

Next, let's calculate the time it takes for the rock to sink from the water's surface to the bottom of the lake.

Since the rock floats gently down through the water at a constant speed of 1.4 m/s, we can use the equation:

t = d / v

Where t is the time, d is the distance, and v is the velocity.

The distance the rock needs to sink is the sum of the tree height and the lake depth:

d = 22 + 4
d = 26 m

Substituting the values:

t = 26 / 1.4
t ≈ 18.57 seconds

So, it takes approximately 18.57 seconds for the rock to sink from the water's surface to the bottom of the lake.

Finally, the total elapsed time is the sum of the time it takes to fall from the tree to the water's surface and the time it takes to sink from the water's surface to the bottom of the lake:

total elapsed time = time to fall + time to sink
total elapsed time ≈ 2.01 seconds + 18.57 seconds
total elapsed time ≈ 20.58 seconds

Therefore, the total elapsed time is approximately 20.58 seconds.