The AM-HM Inequality in Average Speed Calculations

Problem: The AM-HM Inequality in Average Speed Calculations

In kinematics, the concept of average speed is frequently misunderstood as the simple arithmetic mean of speeds. This problem examines a vehicle traveling from point A to point B, covering the first half of the total distance with speed ##v_1## and the second half with speed ##v_2##.

(a) Derive the expression for the average speed ##v_{\text{avg}}## over the entire journey.

(b) Using the Arithmetic Mean-Harmonic Mean (AM-HM) inequality, prove that the average speed ##v_{\text{avg}}## is strictly less than the arithmetic mean of the two speeds, ##v_{\text{mean}} = \dfrac{v_1 + v_2}{2}##, provided ##v_1 \neq v_2##.

Worked Solution & Step-by-Step Explanation

Part (a): Deriving the Average Speed

To determine the average speed, we must adhere to the fundamental definition:

Let the total distance of the journey be ##2d##. Consequently, the distance for each segment is ##d##. We define the time taken for each half as follows:

1. Time for the first half: ##t_1 = \dfrac{d}{v_1}##

2. Time for the second half: ##t_2 = \dfrac{d}{v_2}##

The total time ##t_{\text{total}}## is the sum of these intervals:

Factoring out the distance ##d##, we obtain:

Substituting this into the definition of average speed:

This result is the **Harmonic Mean (HM)** of the two speeds.

Part (b): Applying the AM-HM Inequality

The Arithmetic Mean-Harmonic Mean inequality states that for any set of positive real numbers, the arithmetic mean is greater than or equal to the harmonic mean. For two speeds ##v_1## and ##v_2##:

The inequality theorem states:

Substituting our derived ##v_{\text{avg}}## into the inequality:

To prove the inequality is strict when ##v_1 \neq v_2##, consider the difference between the two means:

Since ##v_1, v_2 > 0## and ##(v_1 - v_2)^2 > 0## for all ##v_1 \neq v_2##, the expression is strictly positive. Thus, ##\text{AM} > \text{HM}##, proving that the average speed is strictly less than the arithmetic mean of the speeds in any non-uniform motion over equal distances.

Conceptual Significance for JEE and NEET

Students often err by calculating average speed as ##\dfrac{v_1 + v_2}{2}##. This calculation is only valid if the vehicle travels at speed ##v_1## for a specific *time* duration equal to the time it travels at speed ##v_2##. When distances are equal, the harmonic mean is the only physically correct approach.

**Key Takeaways:**

* **Dimensional Consistency:** Both ##v_{\text{avg}}## and ##v_{\text{mean}}## have dimensions of ##[LT^{-1}]##.

* **Physics vs. Math:** In physical equations, the choice of mean depends on the constraint (equal distance vs. equal time).

* **JEE/NEET Application:** Always identify whether the problem dictates equal time or equal distance before selecting the averaging method.