A rock is dropped from a tree of height 27 m into a lake (depth 5.1 m) below. After entering the water, the rock floats gently down through the water at a constant speed of 1.7 m/s to the bottom of the lake. What is the total elapsed time?
To find the total elapsed time, we need to calculate the time it takes for the rock to fall from the tree to the surface of the water, plus the time it takes for the rock to sink to the bottom of the lake.
First, let's calculate the time it takes for the rock to fall from the tree to the surface of the water. We can use the equation:
\[ t = \sqrt{\frac{2d}{g}} \]
Where:
t = time taken to fall
d = distance fallen (height of the tree)
The rock falls from a height of 27 m, so \( d = 27 \) m. The acceleration due to gravity, g, is approximately 9.8 m/s².
Plugging in these values into the equation, we get:
\[ t = \sqrt{\frac{2 \times 27}{9.8}} \approx 2.17 \ \text{s} \]
So, it takes the rock approximately 2.17 seconds to fall from the tree to the surface of the water.
Next, let's calculate the time it takes for the rock to sink from the surface of the water to the bottom of the lake. The rock sinks at a constant speed of 1.7 m/s, and the depth of the lake is 5.1 m. We can use the equation:
\[ t = \frac{d}{v} \]
Where:
t = time taken to sink
d = depth of the lake
v = sinking speed of the rock
Plugging in these values into the equation, we get:
\[ t = \frac{5.1}{1.7} = 3 \ \text{s} \]
So, it takes the rock approximately 3 seconds to sink from the surface of the water to the bottom of the lake.
To find the total elapsed time, we add the time taken to fall from the tree to the surface of the water to the time taken to sink from the surface of the water to the bottom of the lake:
\[ \text{Total elapsed time} = 2.17 \ \text{s} + 3 \ \text{s} = 5.17 \ \text{s} \]
Therefore, the total elapsed time is approximately 5.17 seconds.