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Chapter 4: Problem 476

If the water falls from a dam into a turbine wheel \(19.6 \mathrm{~m}\) below, then the velocity of water at the turbine is \(\ldots \ldots\) \(\left(\mathrm{g}=9.8 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(9.8 \mathrm{~m} / \mathrm{s}\) (B) \(19.6 \mathrm{~m} / \mathrm{s}\) (C) \(39.2 \mathrm{~m} / \mathrm{s}\) (D) \(98.0 \mathrm{~m} / \mathrm{s}\)

Short Answer

Expert verified
The velocity of the water at the turbine is approximately \(19.6\) m/s. The correct answer is (B).

Step by step solution

01

Identify the variables

Here, we need to identify the known and unknown variables in the problem. The known variables are the height of the falling water (h) which is 19.6 m, and the acceleration due to gravity (g) which is 9.8 m/s². The unknown variable is the final velocity of water at the turbine (v).
02

Choose the appropriate kinematic equation

Since we are given height and acceleration, and need to find final velocity, we can choose the equation: \(v^2 = u^2 + 2gh\), where u is the initial velocity (0 m/s as the water starts at rest), g is the acceleration due to gravity, and h is the height of the water.
03

Plug in the values

Now, we will substitute the given values into the equation: \(v^2 = 0^2 + 2 (9.8) (19.6)\).
04

Calculate the velocity

By calculating the values, we end up with the equation: \(v^2 = 2 (9.8) (19.6)\), and therefore: \(v^2 = 384.16\). Next, take the square root of both sides to find the final velocity (v): \[v = \sqrt{384.16}\]. Finally, we get the velocity of the water at the turbine: \(v ≈ 19.6\) m/s. Among the given options, (B) is closest to our calculated value: 19.6 m/s.

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