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Chapter 3: Problem 385

Same forces act on two bodies of different mass \(2 \mathrm{~kg}\) and $5 \mathrm{~kg}$ initially at rest. The ratio of times required to acquire same final velocity is (A) \(5: 3\) (B) \(25: 4\) (C) \(4: 25\) (D) \(2: 5\)

Short Answer

Expert verified
The ratio of times required for the two masses to reach the same final velocity is \(2:5\). So the correct answer is (D) \(2: 5\).

Step by step solution

01

Apply Newton's second law of motion to each mass

In this step, we will set up equations for each mass. Based on Newton's second law of motion, \(F = ma\), we have: For mass \(2\mathrm{~kg}\): \(F = m_1a_1\) For mass \(5\mathrm{~kg}\): \(F = m_2a_2\) Since the same force acts on both masses, we can write: \(m_1a_1 = m_2a_2\) Now we can find the ratios of their accelerations: \(\frac{a_1}{a_2} = \frac{m_2}{m_1}\) Plug in the values of the masses: \(\frac{a_1}{a_2} = \frac{5}{2}\)
02

Use the equation for final velocity

To find the time required for both masses to reach the same final velocity, we can use the equation: \(v = u + at\) where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is acceleration, and \(t\) is time. Both bodies are initially at rest, so \(u = 0\). Therefore, the equation becomes: \(v = at\) Now we can find the time for each mass: For mass \(2\mathrm{~kg}\): \(t_1 = \frac{v}{a_1}\) For mass \(5\mathrm{~kg}\): \(t_2 = \frac{v}{a_2}\)
03

Calculate the ratio of times

We are looking for the ratio \(t_1 : t_2\). Divide the equations for \(t_1\) and \(t_2\), we get: \(\frac{t_1}{t_2} = \frac{a_2}{a_1}\) Using the value of \(\frac{a_1}{a_2} = \frac{5}{2}\) from Step 1, we can calculate the ratio of times: \(\frac{t_1}{t_2} = \frac{a_2}{a_1} = \frac{2}{5}\) So the ratio of times required for the two masses to reach the same final velocity is \(2:5\). The correct answer is (D) \(2: 5\).

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