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Chapter 4: Problem 26

If a particle travels \(n\) equal distances with speeds \(v_{1}, v_{2}\), then the average speed \(\bar{V}\) of the particle will be such that (1) \(\bar{V}=\frac{v_{1}+v_{2}+\cdots+v_{n}}{n}\) (2) \(\bar{V}=\frac{n v_{1} v_{2}+v_{n}}{v_{1}+v_{2}+v_{3}+\cdots+v_{n}}\) (3) \(\frac{1}{\bar{V}}=\frac{1}{n}\left(\frac{1}{v_{1}}+\frac{1}{v_{2}}+\cdots+\frac{1}{v_{n}}\right)\) (4) \(\bar{V}=\sqrt{v_{1}{ }^{2}+v_{2}{ }^{2}+\cdots+v_{n}{ }^{2}}\)

Short Answer

Expert verified
The correct answer is option (3).

Step by step solution

01

Understanding the Average Speed

The average speed for a particle traveling equal distances with different speeds is calculated using the harmonic mean. This is because the time taken is a key factor in determining average speed, and the particle covers equal distances at different rates (speeds). Thus, we should use the formula for harmonic mean.
02

Applying the Harmonic Mean Formula

The formula for average speed when traveling equal distances at different speeds is \( \bar{V} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \cdots + \frac{1}{v_n}} \). This formula calculates the harmonic mean of the speeds.
03

Identifying the Correct Option

The derived formula \( \bar{V} = \frac{n}{\frac{1}{v_1} + \frac{1}{v_2} + \cdots + \frac{1}{v_n}} \) can be rearranged to match the given option (3): \( \frac{1}{\bar{V}} = \frac{1}{n} \left( \frac{1}{v_1} + \frac{1}{v_2} + \cdots + \frac{1}{v_n} \right) \). Thus, the correct option is (3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Harmonic Mean
The concept of the harmonic mean is crucial when calculating average speeds, especially when distances remain constant and speeds vary. Unlike the arithmetic mean, the harmonic mean takes into account the time spent traveling at each speed. This makes it more accurate for scenarios involving consistent segments of distance. The formula for the harmonic mean of two speeds, for instance, is expressed as \(\bar{V} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}.\)Using the harmonic mean ensures that each speed's contribution is inversely weighted by the time taken at that speed. Thus, faster speeds over shorter times have a larger influence on the average. Applying this when distances are equal offers a precise measure of performance over a multi-speed journey.
Equal Distances
When a particle travels through equal distances, the calculation of average speed requires special attention. In this scenario, equal distances imply that each segment of the journey covers the same mileage but might be completed at different speeds. Since time varies for each segment based on speed, one cannot simply average out speeds arithmetically. Instead, the time factor must be integrated for accuracy. This makes using the harmonic mean necessary and correct, as it accounts for differing time spans across equal distance segments, leading to an accurate computation of the average speed.
Traveling Speeds
Understanding the behavior of traveling speeds in physics involves recognizing how changes in speed impact travel time over the same distance. When speeds are unequal, the time taken for each segment varies, affecting the overall trip duration. Recognizing these different travel scenarios allows for a more in-depth understanding of average speed calculations. Each speed in a journey contributes differently. Faster speeds take less time, thus contributing more weight, which dictates that we need the harmonic mean to get a true representation of average speed. As a result, resolving complex speed variations and their effect on overall travel time becomes a nuanced process.
Problem Solving in Physics
Tackling problems in physics requires a methodological approach to effectively break down concepts and find accurate solutions. Understanding the type of mean to use when calculating average speed is an example of such strategic thinking. The process starts with determining what remains constant鈥� like distance鈥� and what varies, such as speed or time. Recognizing these elements allows the solver to select appropriate formulas and methods, like the harmonic mean, rather than the arithmetic mean. Applying the correct principles ensures precision in problem solving, thereby enhancing the reliability of one's results.

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Most popular questions from this chapter

A point moves with uniform acceleration and \(v_{1}, v_{2}\), and \(v_{3}\) denote the average velocities in the three successive intervals of time \(t_{1}, t_{2}\), and \(t_{3}\). Which of the following relations is correct? (1) \(\left(v_{1}-v_{2}\right):\left(v_{2}-v_{3}\right)=\left(t_{1}-t_{2}\right):\left(t_{2}+t_{3}\right)\) (2) \(\left(v_{1}-v_{2}\right):\left(v_{2}-v_{3}\right)=\left(t_{1}+t_{2}\right):\left(t_{2}+t_{3}\right)\) (3) \(\left(v_{1}-v_{2}\right):\left(v_{2}-v_{3}\right)=\left(t_{1}-t_{2}\right):\left(t_{1}-t_{3}\right)\) (4) \(\left(v_{1}-v_{2}\right):\left(v_{2}-v_{3}\right)=\left(t_{1}-t_{2}\right):\left(t_{2}-t_{3}\right)\)

A particle is moving along the \(x\)-axis whose position given by \(x=4-9 t+\frac{t^{3}}{3}\). Mark the correct statement(s) relation to its motion. (1) The direction of motion is not changing at any of th. instants. (2) The direction of the motion is changing at \(t=3 \mathrm{~s}\). (3) For \(0

A block slides down a smooth inclined plane when released from the top, while another falls freely from the same point. Which of the following is/are correct? (1) Sliding block will reach the ground first (2) Freely falling block will reach the ground first (3) Both the blocks will reach the ground with different speeds (4) Both the blocks will reach the ground with same speed

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