you drop a coin into a fountain from a height of 15 feet. write an equation that's models the height h (in feet) of the coin above the fountain t seconds after it has been dropped. How long is the coin in the air?

To model the height of the coin above the fountain, we can use the equation for the motion of an object under gravity. The general equation for the height \( h(t) \) of an object in free fall is:

\[
h(t) = h_0 - \frac{1}{2} g t^2
\]

Where:
- \( h_0 \) is the initial height (in feet),
- \( g \) is the acceleration due to gravity (in feet per second squared),
- \( t \) is the time in seconds.

Given:
- The initial height \( h_0 = 15 \) feet,
- The acceleration due to gravity \( g = 32 \, \text{ft/s}^2 \) (since we're using feet and seconds).

The equation becomes:

\[
h(t) = 15 - 16t^2
\]

### How long is the coin in the air?
The coin hits the water when its height is 0. Set \( h(t) = 0 \) and solve for \( t \):

\[
0 = 15 - 16t^2
\]

Solving for \( t \):

\[
16t^2 = 15 \quad \Rightarrow \quad t^2 = \frac{15}{16} \quad \Rightarrow \quad t = \sqrt{\frac{15}{16}} \quad \Rightarrow \quad t = \frac{\sqrt{15}}{4}
\]

Approximating the value:

\[
t \approx 0.968 \, \text{seconds}
\]

So, the coin is in the air for approximately **0.968 seconds**.

Would you like further details or have any questions?

### Related Questions:
1. How would the equation change if the coin were thrown upwards instead of dropped?
2. What happens to the time if the height were doubled?
3. How does the acceleration due to gravity affect the fall time?
4. How can you solve the equation if air resistance were considered?
5. What is the coin’s speed when it hits the water?

#### Tip:
Always double-check units when working with equations of motion to ensure they are consistent throughout.

You drop a coin into a fountain from a height of 15 feet. Write an equation that models the height h (in feet) of the coin above the fountain t seconds after it has been dropped. How long is the coin in the air?

Discover how to model the height of a coin dropped from 15 feet using quadratic equations and the laws of motion under gravity. Learn the time the coin stays in the air, applying key concepts from kinematics and free fall physics. Perfect for students in Grades 9-10.

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