2.76. Egg Drop. You are on the roof of the physics building, 46.0 m above the ground (Fig. 2.49). Your physics profes- sor, who is 1.80 m tall, is walk- ing alongside the building at a constant speed of 1.20 m/s. If you wish to drop an egg on your professor's head, where should the professor be when you release the egg? Assume that the egg is in free fall.

The answer is 3.6 m/s how?

To solve this problem, we need to determine how far the professor will walk during the time it takes for the egg to fall from the roof to the professor's head.

### Step 1: Calculate the time it takes for the egg to fall.

The egg is dropped from a height of 46.0 m above the ground. However, the professor's head is 1.80 m tall, so the effective height from which the egg needs to fall is:

\[
\text{Height of fall} = 46.0 \, \text{m} - 1.80 \, \text{m} = 44.2 \, \text{m}
\]

We'll use the following kinematic equation to find the time \( t \) it takes for the egg to fall:

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

Where:
- \( h = 44.2 \, \text{m} \) is the height of fall,
- \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity,
- \( t \) is the time in seconds.

Rearranging the equation to solve for \( t \):

\[
t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 44.2 \, \text{m}}{9.8 \, \text{m/s}^2}}
\]

Let's calculate that.

\[
t = \sqrt{\frac{88.4}{9.8}} = \sqrt{9.02} \approx 3.00 \, \text{seconds}
\]

### Step 2: Calculate the distance the professor will walk.

Since the professor is walking at a constant speed of \( 1.20 \, \text{m/s} \), the distance \( d \) the professor covers in time \( t \) is:

\[
d = v \times t
\]

Substitute the values:

\[
d = 1.20 \, \text{m/s} \times 3.00 \, \text{s} = 3.60 \, \text{m}
\]

### Conclusion

The professor should be **3.60 meters** away from the point directly below where you drop the egg when you release it. The answer of "3.6 m" refers to this distance, not a speed. The speed of 1.20 m/s is the professor's walking speed.

If you have any questions or need further clarification, feel free to ask!

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### Follow-Up Questions:
1. What would happen if the professor walked faster or slower?
2. How would the problem change if the height of the roof were different?
3. What would be the impact of air resistance on this problem?
4. How does the time of fall change with different heights?
5. What other factors could affect the accuracy of the egg hitting the professor's head?

### Tip:
Remember that in free fall problems, the time it takes for an object to fall only depends on the height from which it falls and the acceleration due to gravity.

Explore how to calculate the distance a professor should walk to catch an egg dropped from a physics building roof. This problem involves kinematic equations for free fall and is suitable for high school physics students.

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