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Chapter 2: Problem 72

In a fancy hotel, the back of the elevator is made of glass so that you can enjoy a lovely view on your ride. The elevator travels at an average speed of \(1.75 \mathrm{~m} / \mathrm{s}\). A boy on the 15th floor, \(80.0 \mathrm{~m}\) above the ground level, drops a rock at the same instant the elevator starts its ascent from the 1st to the 5th floor. Assume the elevator travels at its average speed for the entire trip and neglect the dimensions of the elevator. a) How long after it was dropped do you see the rock? b) How long does it take for the rock to reach ground level?

Short Answer

Expert verified
b) What is the time it takes for the rock to reach the ground level?

Step by step solution

01

Calculate initial height difference

We are given that the boy is on the 15th floor, which is \(80.0 \mathrm{~m}\). The elevator is at the ground level, so the initial height difference is: \(h_{initial} = 80.0 \mathrm{~m}\) **Step 2: Calculate the moment the rock becomes visible to the elevator**
02

Determine when the rock is visible

For the rock to become visible to the elevator, we have to equate their heights: \(h_{elevator} = h_{rock}\) \(v_{elevator} \cdot t_{visible} = 80.0 \mathrm{~m} - \frac{1}{2}g \cdot t_{visible}^2\) Plugging the given value of \(v_{elevator} = 1.75 \mathrm{~m} / \mathrm{s}\), and \(g = 9.8 \mathrm{~m} / \mathrm{s}^2\), we get: \(1.75 \mathrm{~m} / \mathrm{s} \cdot t_{visible} = 80 \mathrm{~m} - \frac{1}{2}(9.8 \mathrm{~m} / \mathrm{s}^2) \cdot t_{visible}^2\) Carefully solve this equation for \(t_{visible}\), which will give you the time after which the rock becomes visible to the elevator. **Step 3: Calculate the time it takes for the rock to reach ground level**
03

Determine the time for the rock to reach ground level

To find the time it takes for the rock to reach the ground level, we use the equation of motion for the rock: \(d_{rock} = 80.0\mathrm{~m} - \frac{1}{2}g \cdot t_{ground}^2\) Plugging the values, we get: \(0 = 80\mathrm{~m} - \frac{1}{2}(9.8\mathrm{~m} / \mathrm{s}^2) \cdot t_{ground}^2\) Carefully solve this equation for \(t_{ground}\), which will give you the time it takes for the rock to reach ground level.

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