To find the speed of the boat, we need to solve this problem in two main parts: the time it takes for the key to fall, and then use that time to find the boat's speed.

Step 1: Time it takes for the key to fall

The key is falling under the influence of gravity. The vertical motion of the key follows the equation for free fall:

$y = \frac{1}{2} g t^2$

Where:

• $y = 47 \, \text{m}$ is the height from which the key is dropped,
• $g = 9.8 \, \text{m/s}^2$ is the acceleration due to gravity,
• $t$ is the time it takes for the key to reach the water.

Rearranging the equation to solve for $t$:

$t = \sqrt{\frac{2y}{g}}$

Substituting the values:

$t = \sqrt{\frac{2 \times 47}{9.8}} = \sqrt{\frac{94}{9.8}} \approx \sqrt{9.59} \approx 3.1 \, \text{seconds}$

So, it takes about 3.1 seconds for the key to fall.

Step 2: Calculate the speed of the boat

The boat is 18 meters away from the point where the key is released, and it moves at a constant speed to reach the point where the key falls. The boat covers 18 meters in the same time it takes the key to fall, which is 3.1 seconds.

To find the speed of the boat, we use the formula for speed:

$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$

Substituting the known values:

$\text{Speed} = \frac{18}{3.1} \approx 5.8 \, \text{m/s}$

Thus, the speed of the boat is approximately 5.8 m/s.

---

Would you like further clarification or more details on any part of the solution?

Here are some related questions for exploration:

1. How would the boat's speed change if the key were dropped from a different height?
2. What would happen if the boat's velocity were not constant, but instead accelerated?
3. How could we apply this same approach to a projectile thrown horizontally instead of a falling key?
4. What assumptions are made in this calculation about the motion of the boat and key?
5. How would air resistance affect the fall of the key and the boat's motion?

Tip: Always break down motion problems into separate horizontal and vertical components. The vertical motion (gravity) is often the key to solving these problems when the horizontal motion (boat's speed) depends on the time taken for the fall.