Solved Problem on One-dimensional Motion

A car in a straight road does half the route with an average speed of v₁ and the other half with an average speed of v₂. Determine the average speed vm of the whole travel.

Problem data:

Average speed of the car in the first half of the route: v₁;
Average speed of the car in the second half of the route: v₂.

Problem diagram:

We will name each half of the route ΔS, and the times spent in each half of the route Δt₁ and Δt₂ (Figure 1).

Figure 1

Solution

As each half of the route is ΔS, the total route will be

\[ \Delta S_{t}=\Delta S+\Delta S=2\Delta S \]

We have to find the time the car takes to travel in each half of the road, from the expression for the average speed

\[ \begin{gather} \bbox[#99CCFF,10px] {v_{m}=\frac{\Delta S}{\Delta t}=\frac{S_{f}-S_{i}}{t_{f}-t_{i}}} \tag{I} \end{gather} \]

writing this equation for each half of the road

\[ \begin{gather} \Delta t_{1}=\frac{\Delta S}{v_{1}} \tag{II-a} \end{gather} \]

\[ \begin{gather} \Delta t_{2}=\frac{\Delta S}{v_{2}} \tag{II-b} \end{gather} \]

The total time Δt of the trip will be the sum of the expressions (II-a) and (II-b), we can write the expression (I) for the average speed of the whole road.

\[ \begin{gather} v_{m}=\frac{\Delta S_{t}}{\Delta t}\\[8pt] v_{m}=\frac{\Delta S_{t}}{\Delta t_{1}+\Delta t_{2}}\\[8pt] v_{m}=\frac{2\Delta S}{\dfrac{\Delta S}{v_{1}}+\dfrac{\Delta S}{v_{2}}} \end{gather} \]

the common factor among the fractions of the denominator will be \( v_{1}v_{2} \)

\[ v_{m}=\frac{2\Delta S}{\dfrac{\Delta S v_{2}+\Delta S v_{1}}{v_{1}v_{2}}} \]

factoring the ΔS factor in the numerator and denominator and turning the denominator upside down

\[ \begin{gather} v_{m}=\frac{\cancel{\Delta S}}{\cancel{\Delta S}}\frac{2}{\dfrac{v_{2}+v_{1}}{v_{1}v_{2}}}\\[5pt] v_{m}=\frac{2v_{1}v_{2}}{(v_{1}+v_{2})} \end{gather} \]

the total average speed will be

\[ \bbox[#FFCCCC,10px] {v_{m}=\frac{2v_{1}.v_{2}}{v_{1}+v_{2}}} \]

Note: This type of average is called the Harmonic Mean, in this case of 2 numbers. Contrary to what many would expect, the average is not an Arithmetic Mean

\[ V_{m}=\frac{v_{1}+v_{2}}{2} \]

If the car ran through the road, with half the total time of the trip (and not half the space) in the first part, with an average speed v₁, and in the second part of the trip with the other half of the total time, with an average speed v₂, then it would be validated the Arithmetic Mean.

Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .