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The problem describes a scenario where a vase falls from a window 12.7 meters above the ground with an initial downward velocity of 5.2 m/s. Let's solve each part step-by-step.

### Given:
- Height, \( h = 12.7 \) meters
- Initial velocity, \( v_0 = 5.2 \) m/s (downward)
- Acceleration due to gravity, \( g = 9.8 \) m/s\(^2\)

### a) Final Velocity before hitting the ground

To find the final velocity of the vase just before it hits the ground, we can use the kinematic equation:

\[
v^2 = v_0^2 + 2gh
\]

where:
- \( v \) is the final velocity,
- \( v_0 = 5.2 \) m/s is the initial velocity,
- \( g = 9.8 \) m/s\(^2\) is the acceleration due to gravity,
- \( h = 12.7 \) m is the height.

Substitute the values:

\[
v^2 = (5.2)^2 + 2 \times 9.8 \times 12.7
\]

Let's calculate this.

### b) Time to reach the ground

To find the time it takes for the vase to reach the ground, we can use the following equation:

\[
h = v_0 t + \frac{1}{2}gt^2
\]

Rearranging for \( t \):

\[
0 = \frac{1}{2}gt^2 + v_0t - h
\]

Plugging in the values:

\[
0 = \frac{1}{2} \times 9.8 t^2 + 5.2t - 12.7
\]

We can solve this quadratic equation for \( t \).

### c) Velocity after 3.5 seconds

To find the velocity of the vase after 3.5 seconds, use the formula:

\[
v = v_0 + gt
\]

Substitute the values:

\[
v = 5.2 + 9.8 \times 3.5
\]

Let's compute all these values now.Here are the solutions to the problem:

### a) Final Velocity Before Reaching the Ground
The final velocity of the vase just before it hits the ground is approximately **16.61 m/s**.

### b) Time to Reach the Ground
The time it takes for the vase to reach the ground is approximately **1.16 seconds**.

### c) Velocity After 3.5 Seconds
The velocity of the vase after 3.5 seconds is **39.5 m/s**.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. How would the results change if the initial velocity was upward instead of downward?
2. What is the kinetic energy of the vase just before it hits the ground?
3. How does air resistance affect the final velocity and time to reach the ground?
4. If the height was doubled, what would be the new time to reach the ground?
5. What is the maximum height the vase would reach if thrown upwards with a given initial velocity?

**Tip:** When solving problems involving free fall, always keep track of the direction of velocity and acceleration.

A dog is running around a room in a condominium. As it was running around the room, it accidentally hit a vase placed by the window. If the window is 12.7 m above the ground, solve for the following assuming the dog hit the vase with a downward velocity of 5.2 m/s:

- What is the final velocity before it exactly reaches the ground?
- How long does it take the vase to reach the ground?
- After 3.5 s, what is the velocity of the vase during the fall?

This problem explores a scenario where a vase is dropped from a height of 12.7 meters with an initial velocity of 5.2 m/s downward. Learn how to calculate the final velocity, the time it takes to reach the ground, and the velocity at a specific time point during free fall. Perfect for high school students learning kinematics.

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