Calculating Mean Speed for a Multi-Segment Trip

The problem asks us to find the mean speed of a car journey where the speed varies over different time intervals. To find the mean speed for the entire trip, we need to calculate the total distance covered and divide it by the total time taken.

The formula for mean speed is:

$$ \text{Mean Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$

The trip is divided into three segments, each with a different driver, duration, and average speed.

• Segment 1: P drives for 1 hour at 40 km/h.
• Segment 2: Q drives for 2 hours at 50 km/h.
• Segment 3: R drives for 3 hours at 60 km/h.

The total duration of the trip is $1 \text{ hour} + 2 \text{ hours} + 3 \text{ hours} = 6 \text{ hours}$.

Step-by-Step Calculation of Distances

We calculate the distance covered in each segment using the formula: Distance = Speed $\times$ Time.

• Distance covered by P:
  Speed = 40 km/h
  Time = 1 hour
  Distance$_{P}$ = $40 \text{ km/h} \times 1 \text{ hour} = 40 \text{ km}$
• Distance covered by Q:
  Speed = 50 km/h
  Time = 2 hours
  Distance$_{Q}$ = $50 \text{ km/h} \times 2 \text{ hours} = 100 \text{ km}$
• Distance covered by R:
  Speed = 60 km/h
  Time = 3 hours
  Distance$_{R}$ = $60 \text{ km/h} \times 3 \text{ hours} = 180 \text{ km}$

Calculating Total Distance

The total distance is the sum of the distances covered in each segment:

$$ \text{Total Distance} = \text{Distance}_P + \text{Distance}_Q + \text{Distance}_R $$

$$ \text{Total Distance} = 40 \text{ km} + 100 \text{ km} + 180 \text{ km} = 320 \text{ km} $$

Calculating Mean Speed

Now we can calculate the mean speed using the total distance and total time:

$$ \text{Mean Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$

$$ \text{Mean Speed} = \frac{320 \text{ km}}{6 \text{ hours}} $$

Performing the division:

$$ \text{Mean Speed} = \frac{320}{6} \text{ km/h} = \frac{160}{3} \text{ km/h} $$

To find the approximate value:

$$ \frac{160}{3} \approx 53.333... \text{ km/h} $$

The mean speed of the trip is approximately 53.33 km/h.

Comparing with Options

Let's compare our calculated mean speed with the given options:

• Option 1: 50.23 km/h
• Option 2: 61.35 km/h
• Option 3: 45.25 km/h
• Option 4: 53.33 km/h

Our calculated value, approximately 53.33 km/h, matches Option 4.

Revision Table: Speed, Distance, and Time

Additional Information: Calculating Average Speed

Calculating the average speed is crucial in problems involving journeys over different segments or varying speeds. It's important not to simply average the speeds of the individual segments unless the time intervals or distances covered in each segment are equal. In this problem, the time intervals are different (1 hour, 2 hours, 3 hours), so we must use the total distance and total time approach.

Consider a case where speed changes over equal time intervals. If a car travels at speed $v_1$ for time $t$ and then at speed $v_2$ for the same time $t$, the total distance is $v_1 t + v_2 t$. The total time is $t+t=2t$. The average speed is $\frac{v_1 t + v_2 t}{2t} = \frac{t(v_1+v_2)}{2t} = \frac{v_1+v_2}{2}$. In this specific case of equal time intervals, the average speed is the simple arithmetic mean of the speeds. However, this shortcut doesn't apply when time intervals are unequal, as shown in the problem solved above.

Similarly, if speed changes over equal distances. If a car travels distance $d$ at speed $v_1$ and then the same distance $d$ at speed $v_2$. The time taken for the first part is $t_1 = d/v_1$ and for the second part is $t_2 = d/v_2$. The total distance is $d+d=2d$. The total time is $t_1+t_2 = d/v_1 + d/v_2 = d(\frac{1}{v_1} + \frac{1}{v_2}) = d(\frac{v_1+v_2}{v_1 v_2})$. The average speed is $\frac{2d}{d(\frac{v_1+v_2}{v_1 v_2})} = \frac{2v_1 v_2}{v_1+v_2}$. This is the harmonic mean of the two speeds.

This problem highlights that calculating average speed requires careful consideration of the total distance covered over the total duration of the journey.