One of your fellow students drops their empty drink bottle in the trash can at the front of the classroom. The bottle takes 0.64s to travel from the student's hand to the bottom of the trash can. How far did the bottle travel during this time?      


To determine how far the bottle traveled during the 0.64 seconds, we can use the following kinematic equation for free-falling objects under gravity:

\[
d = \frac{1}{2} g t^2
\]

Where:
- \( d \) is the distance the object falls (in meters),
- \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)),
- \( t \) is the time of fall (0.64 seconds).

Substituting the values:

\[
d = \frac{1}{2} \times 9.8 \, \text{m/s}^2 \times (0.64 \, \text{s})^2
\]

Now, solving this:

\[
d = \frac{1}{2} \times 9.8 \times 0.4096
\]
\[
d = 4.9 \times 0.4096
\]
\[
d = 2.006 \, \text{meters}
\]

Thus, the bottle traveled approximately **2.01 meters**.

Would you like more details or further clarification?

Here are 5 related questions to expand the concept:
1. How would the distance change if the bottle was thrown upward instead of dropped?
2. What would be the time of travel if the trash can were 3 meters tall?
3. How would the calculation change on a planet with a different gravitational acceleration?
4. Can you derive the formula used for vertical motion under constant acceleration?
5. What role does air resistance play in this scenario?

**Tip:** For free-falling objects, the only force acting on them is gravity, provided we neglect air resistance!

One of your fellow students drops their empty drink bottle in the trash can at the front of the classroom. The bottle takes 0.64s to travel from the student's hand to the bottom of the trash can. How far did the bottle travel during this time?

Learn how to calculate the distance traveled by a falling object in free fall using the kinematic equation d = (1/2) * g * t^2. This problem involves kinematics and gravitational acceleration, providing a clear step-by-step solution. Suitable for students in Grades 9-11.

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