/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 130 A vehicle travels half the dista... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 130

A vehicle travels half the distance \(L\) with speed \(V_{1}\) and the other half with speed \(V_{2}\), then its average speed is (A) \(\frac{V_{1}+V_{2}}{2}\) (B) \(\frac{2 V_{1}+V_{2}}{V_{1}+V_{2}}\) (C) \(\frac{2 V_{1} V_{2}}{V_{1}+V_{2}}\) (D) \(\frac{2 V_{1} V_{2}}{V_{1}+V_{2}}\)

Short Answer

Expert verified
The short answer is: The average speed of the vehicle is \(\frac{2 V_{1} V_{2}}{V_{1}+V_{2}}\).

Step by step solution

01

Determine the time taken for each half

To do this, we use the formula for distance: \(Distance = Speed \times Time\). Rearranging the formula to find time, we get: \(Time = \frac{Distance}{Speed}\). For the first half of the distance (\(L/2\)), the speed is \(V_{1}\). Therefore, the time taken for the first half is: \(Time_{1} = \frac{L/2}{V_{1}} = \frac{L}{2 V_{1}}\) For the second half of the distance (\(L/2\)), the speed is \(V_{2}\). Therefore, the time taken for the second half is: \(Time_{2} = \frac{L/2}{V_{2}} = \frac{L}{2 V_{2}}\)
02

Calculate the total time taken

To find the total time for the entire distance, we add the times taken for each half: \(Total ~Time = Time_{1} + Time_{2}\) Substitute the values of \(Time_{1}\) and \(Time_{2}\) from step 1: \(Total ~Time = \frac{L}{2 V_{1}} + \frac{L}{2 V_{2}}\)
03

Calculate the average speed

To find the average speed, we divide the total distance by the total time: \(Average ~Speed = \frac{Total ~Distance}{Total ~Time}\) The total distance covered is \(L/2+L/2 = L\). Substitute the total distance and total time found in step 2: \(Average ~Speed = \frac{L}{\left(\frac{L}{2 V_{1}} + \frac{L}{2 V_{2}}\right)}\)
04

Simplify the equation

Simplify the equation to find the average speed: \(Average ~Speed = \frac{L}{L\left(\frac{1}{2 V_{1}} + \frac{1}{2 V_{2}}\right)}\) Divide both the numerator and denominator by L: \(Average ~Speed = \frac{1}{\left(\frac{1}{2 V_{1}} + \frac{1}{2 V_{2}}\right)}\) We can simplify further by finding the common denominator in the brackets: \(Average ~Speed = \frac{1}{\left(\frac{V_{2} + V_{1}}{2 V_{1} V_{2}}\right)}\) Now, to simplify, we can invert the fraction inside the brackets and multiply it with the numerator: \(Average ~Speed = \frac{2 V_{1} V_{2}}{V_{1}+V_{2}}\) Comparing our result with the options given, we find the correct answer is: (C) \(\frac{2 V_{1} V_{2}}{V_{1}+V_{2}}\)

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