You throw a small rock straight up from the edge of a highway bridge that crosses a river. The rock passes you on its way down, 5.00 ss after it was thrown. What is the speed of the rock just before it reaches the water 27.0 mm below the point where the rock left your hand? Ignore air resistance.

To solve this problem, let's break it down into steps. We'll use the principles of kinematics to determine the speed of the rock just before it reaches the water.

### Given:
- The total time for the rock to return to the starting point: \( t = 5.00 \) seconds
- The distance from the starting point to the water: \( h = 27.0 \) meters
- Acceleration due to gravity: \( g = 9.81 \) m/s²
- Initial velocity: \( v_0 \) (to be determined)

### Step 1: Find the initial velocity (\( v_0 \))

When the rock returns to the same height from which it was thrown after 5.00 seconds, the time it takes to reach the highest point (where the velocity is 0) is half of that, i.e., \( t_{up} = \frac{5.00}{2} = 2.50 \) seconds.

The initial velocity can be found using the equation:
\[
v_0 = g \times t_{up}
\]

### Step 2: Find the speed of the rock just before it reaches the water

The rock will travel an additional 27.0 meters downward after returning to its original height. To find the speed just before it hits the water, we can use the following kinematic equation:
\[
v^2 = v_0^2 + 2gh
\]
where:
- \( v_0 \) is the velocity of the rock when it reaches the starting point again (which will be the same as the initial velocity due to symmetry),
- \( g \) is the acceleration due to gravity, and
- \( h \) is the height.

Let's calculate the initial velocity \( v_0 \) and then use it to find the final speed \( v \) just before the rock hits the water.The initial velocity of the rock when it was thrown upward is approximately \( 24.53 \) m/s.

The speed of the rock just before it reaches the water, 27.0 meters below the point where it was thrown, is approximately \( 33.63 \) m/s.

Would you like further details or explanations on any part of the solution? Here are some related questions to consider:

1. What would happen to the final speed if the height was doubled?
2. How would air resistance affect the final speed?
3. Can you derive a general formula for the final speed given any height?
4. What would the initial velocity need to be for the rock to take 10 seconds to return to the same height?
5. How would the motion differ if the rock was thrown downward instead of upward?

**Tip:** When dealing with vertical motion, remember that the initial velocity and the acceleration due to gravity both affect the time it takes to reach the maximum height.

Learn how to calculate the speed of a rock in a projectile motion problem involving vertical motion and gravity. This problem explores kinematics principles, including equations for initial and final velocity and the impact of gravity on motion. Suitable for advanced high school physics students.

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