Andrea drops an apple from the empire state building. How far has it fallen after 8 seconds?

To calculate how far an object falls under gravity after a certain time, we use the formula:

\[
d = \frac{1}{2}gt^2
\]

where:
- \( d \) is the distance fallen,
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \)),
- \( t \) is the time in seconds.

Given \( t = 8 \) seconds, let's substitute the values:

\[
d = \frac{1}{2} \cdot 9.8 \cdot 8^2
\]
\[
d = \frac{1}{2} \cdot 9.8 \cdot 64
\]
\[
d = \frac{1}{2} \cdot 627.2
\]
\[
d = 313.6 \, \text{meters}
\]

So, after 8 seconds, the apple has fallen **313.6 meters**.

Would you like a more detailed explanation on any part of this? 

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### Further Questions
1. How far has the apple fallen after 5 seconds?
2. How does air resistance affect the distance fallen?
3. What would the speed of the apple be after 8 seconds?
4. How long would it take for the apple to reach the ground if dropped from a height of 500 meters?
5. What distance will the apple fall in the first 2 seconds?

### Tip
Remember that in free-fall motion (without air resistance), distance increases with the square of time, meaning objects fall faster the longer they fall.

Learn how to calculate the distance an apple falls in free fall under gravity after 8 seconds using the formula d = 0.5 * g * t^2. With an acceleration due to gravity of 9.8 m/s², the apple falls a total of 313.6 meters. This solution is suitable for students in Grades 9-12 studying kinematics and free-fall motion.

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