To solve this problem, we'll determine the time it takes for the egg to fall from the roof to the professor's head, and then use that time to calculate the distance the professor should be from the point directly below where the egg is dropped.

Step 1: Calculate the time of free fall

The egg is dropped from a height of $h = 46.0 \, \text{m}$ above the ground, and the professor's height is $h_p = 1.80 \, \text{m}$. Therefore, the egg needs to fall a distance of: $h_f = 46.0 \, \text{m} - 1.80 \, \text{m} = 44.2 \, \text{m}$

We can use the kinematic equation for free fall: $h_f = \frac{1}{2} g t^2$ where:

• $h_f = 44.2 \, \text{m}$
• $g = 9.81 \, \text{m/s}^2$ (acceleration due to gravity)
• $t$ is the time in seconds

Solving for $t$: $t = \sqrt{\frac{2 h_f}{g}} = \sqrt{\frac{2 \times 44.2 \, \text{m}}{9.81 \, \text{m/s}^2}} \approx \sqrt{9.01 \, \text{s}^2} \approx 3.00 \, \text{s}$

Step 2: Determine the position of the professor

The professor walks at a constant speed of $v_p = 1.20 \, \text{m/s}$. The distance the professor will travel in the time $t$ is: $d = v_p \times t = 1.20 \, \text{m/s} \times 3.00 \, \text{s} = 3.60 \, \text{m}$

Conclusion

To hit the professor on the head with the egg, the professor should be 3.60 meters away from the point directly below the spot where the egg is dropped when you release it.

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Would you like a more detailed explanation, or do you have any questions?

Related Questions:

1. What would happen if the professor walked at a different speed?
2. How would the problem change if the height of the building were different?
3. What if the egg were thrown with an initial velocity instead of being dropped?
4. How would the time of fall change if the egg were dropped from a lower height?
5. How would air resistance affect the fall time and the result?

Tip: When solving problems involving free fall, always start by calculating the time of descent, which you can then use to determine other factors like horizontal distance traveled.