Question

A car covers one third part of its straight path with speed \(\mathrm{V}_{1}\) and the rest with speed \(\mathrm{V}_{2}\). What is its average speed? (A) \(\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]\) (B) \(\left[\left(2 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(3 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]\) (C) \(\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(\mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]\) (D) \(\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]\)

Short answer

The short answer is: \(\boxed{\text{(D)}\:\frac{3V_1V_2}{2V_1+2V_2}}\).

Step-by-step solution

Calculate the time taken for each part

Let \(d_1\) be the distance covered with speed \(V_1\), and \(d_2\) be the distance covered with speed \(V_2\). We know that \(d_1 = \frac{1}{3}d\) (where \(d\) is the total distance), and \(d_2 = \frac{2}{3}d\). Now, using the formula of time, \[Time = \frac{Distance}{Speed}\] Calculate the time taken for each part: \[t_1 = \frac{d_1}{V_1} = \frac{\frac{1}{3}d}{V_1}\] \[t_2 = \frac{d_2}{V_2} = \frac{\frac{2}{3}d}{V_2}\]

Calculate Total Time

Add the time taken for each part to calculate the total time: \[Total\:Time = t_1 + t_2 = \frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}\]

Calculate the Average Speed

Now, using the formula for the average speed and the total time calculated in step 2, \[Average\:Speed = \frac{Total\:Distance}{Total\:Time} = \frac{d}{\frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}}\]

Simplify the formula and compare to the given options

\[Average\:Speed = \frac{d}{\frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}} \cdot \frac{3V_1V_2}{3V_1V_2}\] \[Average\:Speed = \frac{3V_1V_2d}{\left(\frac{1}{3}dV_2\right) + \left(\frac{2}{3}dV_1\right)}\] \[Average\:Speed = \frac{3V_1V_2d}{\left(V_2d\right) + \left(2V_1d\right)}\] Since the total distance (d) is the same for both parts, we can cancel it out: \[Average\:Speed = \frac{3V_1V_2}{\left(V_2\right) + \left(2V_1\right)}\] Comparing to the given options, the final answer is (D) \(\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]\).

Key concepts

Distance and Speed Relationship

The relationship between distance and speed is fundamental in understanding motion. When we talk about distance and speed, we mean how far an object travels and how fast it's moving while covering that distance. To determine the time it takes for an object to travel a certain distance, we use the formula:

• Time = Distance / Speed

This formula indicates that an increase in speed results in less time taken to cover the same distance, assuming distance remains constant. Conversely, if the speed decreases, the time taken will increase.
This straightforward relationship helps us calculate the time taken to travel specific segments of a journey, like in the exercise where a car travels parts of its journey at different speeds. Understanding these basics is vital as they apply to various types of motion problems we encounter in physics and daily life. Moreover, when calculating the total time for a journey divided into sections with different speeds, the distances covered need to be assessed individually. This ensures precise motion calculations and helps in solving complex problems like determining average speed.

Motion in One Dimension

Motion in one dimension, often referred to as linear motion, occurs when an object moves along a straight line. This type of motion is particularly simple, as it involves only one spatial coordinate. When analyzing motion in one dimension, we focus on three primary factors:

• Distance travelled
• Speed of the object
• Time taken

An example is a car traveling down a straight road. The motion is straightforward since we only have to consider changes along the road's length, rather than any changes in direction or elevation. In problems involving one-dimensional motion, like the scenario provided, the path is split into sections. Each section has a distinct speed, and understanding how the car transitions between these different speeds over the same straight path helps in computing useful quantities, such as the total time of travel or average speed. With the given conditions, such a straightforward path allows for clear calculations without the complication of additional dimensions, making initial problems and calculations more intuitive and easier to understand.

Time Calculation in Motion Problems

Time calculation in motion problems is crucial when determining how long it takes for an object to complete a journey. Knowing how to calculate time accurately is essential for solving various motion-related problems. For any given segment of a journey, the time it takes is calculated using the basic formula:

• Time = Distance / Speed

This formula is used for each part of a journey if the speed is not constant throughout.In the exercise, the path is divided into two segments, each covered at a different speed (\( V_1 \) and \( V_2 \)). For each part of the journey, the time taken is:

• First part: \( t_1 = \frac{\frac{1}{3}d}{V_1} \)
• Second part: \( t_2 = \frac{\frac{2}{3}d}{V_2} \)

The total time is simply the sum of these individual times:

• Total Time = \( t_1 + t_2 \)

Careful calculation and understanding of the time for each segment allow us to determine the overall average speed of the journey, as seen in the solution. Recognizing the separate time durations for varied speeds is a valuable skill in both the academic study of physics and practical applications.