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Chapter 2: Problem 28

A stone is allowed to fall from the top of a tower and cover half the height of the tower in the last second of its journey. The time taken by the stone to reach the foot of the tower is (A) \((2-\sqrt{2}) \mathrm{s}\) (B) \(4 \mathrm{~s}\) (C) \((2+2 \sqrt{2}) \mathrm{s}\) (D) \((2+\sqrt{2}) \mathrm{s}\)

Short Answer

Expert verified
The total time taken by the stone to reach the foot of the tower is \((2 + \sqrt{2})s\).

Step by step solution

01

Write down the given information

We are given that the stone falls from a tower of height H, and it covers half the height (H/2) in the last second. The initial velocity is 0 m/s as the stone is at rest, and the acceleration due to gravity is 9.8 m/s虏.
02

Calculate time for the last second

Using the first equation, as described in the analysis: \[h = ut + \frac{1}{2}at^2\] We can plug in the values for h, u, and a for the last second to find out the time taken: \[\frac{H}{2} = 0 \times (1) + \frac{1}{2}(9.8)(1)^2\] This simplifies to: \[\frac{H}{2} = 4.9\]
03

Calculate time for the first part of the fall

Now we need to find the time taken for the stone to fall to the halfway point of the tower. Let this time be t. We have the following equation: \[H - \frac{H}{2} = ut + \frac{1}{2}at^2\] The equation becomes: \[\frac{H}{2} = 0 \times t + \frac{1}{2}(9.8)t^2\] Plug in the value of H/2, which is 4.9: \[4.9 = \frac{1}{2}(9.8)t^2\]
04

Solve for t

To solve for t, divide both sides of the equation by 4.9: \[1 = \frac{1}{2}(9.8)t^2\] Now, multiply both sides of the equation by 2 to get: \[2 = (9.8)t^2\] Divide by 9.8: \[\frac{2}{9.8} = t^2\] Then, take the square root of both sides to get the value of t: \[t = \sqrt{\frac{2}{9.8}}\]
05

Calculate the total time

The total time will be the sum of the time it takes to fall to the halfway point and the last second: Total time = t (time for first part) + 1 (last second) Total time = \(\sqrt{\frac{2}{9.8}} + 1\)
06

Simplify and match the answer with the given options

To match the answer with the given options, find the common denominator for both terms and obtain: Total time = \(\frac{\sqrt{2}}{\sqrt{9.8}} + \frac{\sqrt{9.8}}{\sqrt{9.8}}\) Total time = \(\frac{\sqrt{2}+\sqrt{9.8}}{\sqrt{9.8}}\) Now, we can see that our answer matches option (D): Total time = \((2 + \sqrt{2})s\)

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