You are to drive 300 km to an interview. The inter- d, a race car must be driven first in one d

step 1 In this problem of motion along a straight line we have given that you are to drive 300 kilomete

You are to drive 300 km to an interview. The inter- d, a...

You are to drive $300 \mathrm{~km}$ to an interview. The inter- $d,$ a race car must be driven first in one direction (in time $\left.t_{1}\right)$ and then in the opposite direction (in time $t_{2}$ ). (a) To eliminate the effects of the wind and obtain the car's speed $v_{c}$ in a windless situation, should we find the average of $\bar{d} l_{1}$ and $d h_{2}$ (method 1) or should we divide $d$ by the average of $l_{1}$ and $l_{2} ?$ (b) What is the fractional difference in the two methods when a steady wind blows along the car's route and the ratio of the wind speed $v_{n}$ to the car's speed $v_{c}$ is $0.0240 ?$

Key Concepts

Harmonic Mean vs Arithmetic Mean

This concept highlights the difference between averaging rates (such as speed) using the arithmetic mean and the harmonic mean. For journeys of equal distance, the arithmetic mean of the speeds (obtained by measuring each leg of the journey) properly cancels symmetrical errors like those introduced by a constant wind. In contrast, taking the average of the times and then dividing the distance by this average generally corresponds to the harmonic mean, which does not yield the true speed when external factors are at play.

Fractional Error Analysis

Fractional error analysis involves determining the relative difference between two measurement methods or models, typically expressed as a ratio to the true value. In the context of eliminating wind effects in speed measurements, this concept is used to quantify the error introduced by an averaging method that does not fully account for the wind. When the ratio of the wind speed to the object’s true speed is small, the fractional error—the amount by which the estimated speed deviates from the true speed—can be computed, providing insight into the accuracy of the method used.

Relative Velocity

This concept refers to the idea that when an object moves in a medium that is itself moving (such as a car in a windy environment), the effective speed of the object is altered by the medium’s speed. In problems like these, the object's speed relative to the ground is determined by vectorially adding the velocity of the medium (wind) to the object's own speed. This is crucial to understand so that corrections can be made to determine the true speed of the object when the external influence is absent.

Averaging Methods for Speed

When determining an object’s true speed from measurements taken under varying conditions, it is important to choose the appropriate method of averaging. In the context of motion over equal distances, two methods might be considered: taking the arithmetic mean of the speeds measured in opposite directions or computing the speed by dividing the total distance by the average of the times. Understanding which method appropriately cancels out or incorporates external effects—such as wind—is key to obtaining the correct value.

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