A driver increases the speed of his vehicle by 5 km/h after every one hour. If the initial speed of the vehicle was 75 km/h, then what was the average speed of the vehicle in a journey of 4 hours?
82.5 km/h
Calculating Average Speed in a Journey
Let's break down how to calculate the average speed of the vehicle during its 4-hour journey, considering its speed changes each hour.
Understanding the Vehicle's Speed
The problem states the vehicle starts with an initial speed of 75 km/h and increases its speed by 5 km/h every one hour.
Let's list the speed of the vehicle for each hour of the 4-hour journey:
- Hour 1: Initial speed is 75 km/h.
- Hour 2: Speed increases by 5 km/h. Speed becomes $75 + 5 = 80$ km/h.
- Hour 3: Speed increases by another 5 km/h. Speed becomes $80 + 5 = 85$ km/h.
- Hour 4: Speed increases by yet another 5 km/h. Speed becomes $85 + 5 = 90$ km/h.
Calculating Distance Covered Each Hour
The distance covered in each hour can be calculated using the formula: Distance = Speed × Time. Since each interval is exactly one hour, the distance covered in that hour is numerically equal to the speed during that hour (in km).
- Distance in Hour 1: $75 \text{ km/h} \times 1 \text{ h} = 75 \text{ km}$
- Distance in Hour 2: $80 \text{ km/h} \times 1 \text{ h} = 80 \text{ km}$
- Distance in Hour 3: $85 \text{ km/h} \times 1 \text{ h} = 85 \text{ km}$
- Distance in Hour 4: $90 \text{ km/h} \times 1 \text{ h} = 90 \text{ km}$
We can summarize the speed and distance per hour in a table:
| Hour | Speed (km/h) | Time (h) | Distance (km) |
|---|---|---|---|
| 1 | 75 | 1 | 75 |
| 2 | 80 | 1 | 80 |
| 3 | 85 | 1 | 85 |
| 4 | 90 | 1 | 90 |
Calculating Total Distance and Total Time
To find the average speed, we need the total distance covered and the total time taken for the journey.
- Total Distance: Sum of the distances covered in each hour.
Total Distance $= 75 \text{ km} + 80 \text{ km} + 85 \text{ km} + 90 \text{ km}$
Total Distance $= 330 \text{ km}$ - Total Time: The journey lasted for 4 hours, as given in the problem.
Calculating Average Speed
The formula for average speed is:
$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$
Plugging in the values we calculated:
$$\text{Average Speed} = \frac{330 \text{ km}}{4 \text{ h}}$$
Let's perform the division:
$$\text{Average Speed} = 82.5 \text{ km/h}$$
Therefore, the average speed of the vehicle during the 4-hour journey was 82.5 km/h.
Revision Table: Speed and Average Speed
| Concept | Definition | Formula/Calculation |
|---|---|---|
| Initial Speed | Speed at the beginning of the journey. | 75 km/h |
| Speed Increase Rate | How much speed changes per unit time. | +5 km/h every hour |
| Speed per Hour | Speed maintained during each specific hour. | 75, 80, 85, 90 km/h for hours 1, 2, 3, 4 respectively. |
| Total Distance | Sum of distances covered in all parts of the journey. | Sum of distances for each hour: $75+80+85+90 = 330$ km |
| Total Time | Total duration of the journey. | 4 hours |
| Average Speed | Total distance divided by total time. | $\frac{330 \text{ km}}{4 \text{ h}} = 82.5 \text{ km/h}$ |
Additional Information: Understanding Average Speed
Average speed is a crucial concept when dealing with journeys where speed isn't constant. It represents the constant speed at which the vehicle would have to travel to cover the same total distance in the same total time. It is different from the simple average of initial and final speeds if the speed changes in a non-linear way or if the time intervals at different speeds are unequal. In this case, since the speed changes linearly over equal time intervals (1 hour each), the average speed is the arithmetic mean of the speeds at the midpoint of each interval, or calculated accurately using Total Distance / Total Time.
For instance, if the speed was constant at 82.5 km/h for 4 hours, the total distance covered would be $82.5 \text{ km/h} \times 4 \text{ h} = 330 \text{ km}$, which matches the total distance we calculated for the changing speed journey.
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