Speed Calculation for P's Journey

This problem asks us to find the average speed of a car journey undertaken by P. The journey lasts for a total of 5 hours. A key piece of information is that P doubles the speed every hour. We are given the specific speed achieved during the fifth hour, which is 128 km/h. Our goal is to calculate the overall average speed for the entire 5-hour trip.

Calculating Speeds Hour by Hour

To find the average speed, we first need to determine the speed during each individual hour of the journey. Let's represent the speed in the first hour as $S_1$. The problem states that the speed doubles every hour.

• Speed in the 1st hour: $S_1$
• Speed in the 2nd hour: $S_2 = 2 \times S_1$
• Speed in the 3rd hour: $S_3 = 2 \times S_2 = 2 \times (2 \times S_1) = 4 \times S_1$
• Speed in the 4th hour: $S_4 = 2 \times S_3 = 2 \times (4 \times S_1) = 8 \times S_1$
• Speed in the 5th hour: $S_5 = 2 \times S_4 = 2 \times (8 \times S_1) = 16 \times S_1$

We are given that the speed in the fifth hour ($S_5$) is 128 km/h. We can use this information to find the speed in the first hour ($S_1$):

$$16 \times S_1 = 128 \text{ km/h}$$

Solving for $S_1$:

$$S_1 = \frac{128}{16} = 8 \text{ km/h}$$

Now that we know $S_1$, we can list the speed for each hour:

• 1st hour speed: $S_1 = 8$ km/h
• 2nd hour speed: $S_2 = 2 \times 8 = 16$ km/h
• 3rd hour speed: $S_3 = 2 \times 16 = 32$ km/h
• 4th hour speed: $S_4 = 2 \times 32 = 64$ km/h
• 5th hour speed: $S_5 = 2 \times 64 = 128$ km/h

Determining Total Distance Traveled

The distance covered in each hour is calculated using the formula: Distance = Speed $\times$ Time. Since the time interval for each speed is exactly 1 hour, the distance covered in each hour is numerically equal to the speed during that hour.

• Distance in 1st hour ($D_1$) = $8 \text{ km/h} \times 1 \text{ h} = 8$ km
• Distance in 2nd hour ($D_2$) = $16 \text{ km/h} \times 1 \text{ h} = 16$ km
• Distance in 3rd hour ($D_3$) = $32 \text{ km/h} \times 1 \text{ h} = 32$ km
• Distance in 4th hour ($D_4$) = $64 \text{ km/h} \times 1 \text{ h} = 64$ km
• Distance in 5th hour ($D_5$) = $128 \text{ km/h} \times 1 \text{ h} = 128$ km

The total distance ($D_{Total}$) is the sum of the distances covered in all 5 hours:

$$D_{Total} = D_1 + D_2 + D_3 + D_4 + D_5$$

$$D_{Total} = 8 + 16 + 32 + 64 + 128 \text{ km}$$

This sum represents a geometric series. We can calculate the total distance using the sum formula for a geometric series, $S_n = \frac{a(r^n - 1)}{r-1}$, where $a$ is the first term (8 km), $r$ is the common ratio (2), and $n$ is the number of terms (5).

$$D_{Total} = \frac{8(2^5 - 1)}{2-1}$$

$$D_{Total} = \frac{8(32 - 1)}{1}$$

$$D_{Total} = 8 \times 31 = 248 \text{ km}$$

Finding the Average Speed

The average speed is calculated by dividing the total distance traveled by the total time taken for the journey.

The formula for average speed is:

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

In this case, the total distance is 248 km, and the total time is 5 hours.

$$\text{Average Speed} = \frac{248 \text{ km}}{5 \text{ hours}}$$

Performing the division:

$$\text{Average Speed} = 49.6 \text{ km/h}$$

Thus, the average speed of P's journey is 49.6 km/h.