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Chapter 4: Problem 485

If a man increase his speed by \(2 \mathrm{~m} / \mathrm{s}\), his $\mathrm{K} . \mathrm{E}$. is doubled, the original speed of the man is (A) \((2+2 \sqrt{2}) \mathrm{m} / \mathrm{s}\) (B) \((2+\sqrt{2}) \mathrm{m} / \mathrm{s}\) (C) \(4 \mathrm{~m} / \mathrm{s}\) (D) \((1+2 \sqrt{2}) \mathrm{m} / \mathrm{s}\)

Short Answer

Expert verified
The original speed of the man is (B) \((2+\sqrt{2}) \mathrm{m} / \mathrm{s}\).

Step by step solution

01

Recall the formula for Kinetic Energy

Kinetic energy is given by the formula: \(K.E. = \frac{1}{2}mv^2\) where, \(K.E.\) = kinetic energy \(m\) = mass (which remains constant for the man) \(v\) = velocity (speed in this case)
02

Set up the equations

Let the original speed of the man be \(v\) m/s. Then, when the man increases his speed by 2 m/s, the new speed becomes \((v + 2)\) m/s. From the information given, we know that doubling the original kinetic energy is equal to the kinetic energy at the higher speed. So, we can write the equation: \(2 \cdot \frac{1}{2}m(v)^2 = \frac{1}{2}m(v + 2)^2\)
03

Simplify the equation

First, notice that the factors \(\frac{1}{2}m\) can be canceled from both sides: \(2v^2 = (v + 2)^2\) Now, expand and simplify the equation: \(2v^2 = v^2 + 4v + 4\) \(v^2 - 4v - 4 = 0\)
04

Solve the quadratic equation

Now, we must solve the quadratic equation for \(v\). Use the quadratic formula: \(v = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) For the equation \(v^2 - 4v - 4 = 0\), the values are: \(a = 1\) \(b = -4\) \(c = -4\) Plugging these values into the formula: \(v = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(-4)}}{2 (1)}\)
05

Find the value of v

Now, calculate the value of \(v\): \(v = \frac{4 \pm \sqrt{16+16}}{2}\) \(v = \frac{4 \pm \sqrt{32}}{2}\) \(v = 2 \pm \frac{\sqrt{32}}{2}\) The possible values for the original speed are: \(v_1 = 2 + \frac{\sqrt{32}}{2}\) and \(v_2 = 2 - \frac{\sqrt{32}}{2}\) As speed cannot be negative, we can eliminate \(v_2\). Therefore, the original speed of the man is: \(v = 2 + \frac{\sqrt{32}}{2}\) Comparing with the given options, we see that the correct answer is: (B) \((2+\sqrt{2}) \mathrm{m} / \mathrm{s}\)

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Most popular questions from this chapter

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