A journey of 900 km is completed in 11 h. If two-fifth of the journey is completed at the speed of 60 km/h, at what speed (in km/h) is the remaining journey completed?
108
Solving the Journey Speed Problem
This problem involves calculating the speed required for a part of a journey, given the total distance, total time, and details about the initial part of the journey.
Understanding the Problem Details
We are given the following information:
- Total distance of the journey: 900 km
- Total time taken for the journey: 11 hours
- Fraction of the journey completed at a specific speed: two-fifth
- Speed for the first two-fifth of the journey: 60 km/h
We need to find the speed at which the remaining part of the journey was completed.
Step-by-Step Calculation of Journey Segments
Step 1: Calculate the distance of the first part of the journey.
The first part is two-fifth of the total journey.
Distance of the first part = \(\frac{2}{5} \times \text{Total Distance}\)
Distance of the first part = \(\frac{2}{5} \times 900 \text{ km}\)
Distance of the first part = \(2 \times 180 \text{ km}\)
Distance of the first part = 360 km
Step 2: Calculate the time taken for the first part of the journey.
We know the distance of the first part and the speed at which it was covered.
Speed = \(\frac{\text{Distance}}{\text{Time}}\)
Time = \(\frac{\text{Distance}}{\text{Speed}}\)
Time taken for the first part = \(\frac{\text{Distance of the first part}}{\text{Speed in the first part}}\)
Time taken for the first part = \(\frac{360 \text{ km}}{60 \text{ km/h}}\)
Time taken for the first part = 6 hours
Step 3: Calculate the distance of the remaining journey.
The remaining distance is the total distance minus the distance covered in the first part.
Remaining Distance = Total Distance - Distance of the first part
Remaining Distance = 900 km - 360 km
Remaining Distance = 540 km
Step 4: Calculate the time taken for the remaining journey.
The remaining time is the total time for the journey minus the time taken for the first part.
Remaining Time = Total Time - Time taken for the first part
Remaining Time = 11 hours - 6 hours
Remaining Time = 5 hours
Step 5: Calculate the speed for the remaining journey.
We now have the distance and time for the remaining journey. We can calculate the required speed.
Speed = \(\frac{\text{Distance}}{\text{Time}}\)
Speed for the remaining journey = \(\frac{\text{Remaining Distance}}{\text{Remaining Time}}\)
Speed for the remaining journey = \(\frac{540 \text{ km}}{5 \text{ hours}}\)
Speed for the remaining journey = 108 km/h
Summary of Journey Segments
| Segment | Distance | Time Taken | Speed |
|---|---|---|---|
| First Part | 360 km | 6 hours | 60 km/h |
| Remaining Part | 540 km | 5 hours | 108 km/h |
| Total | 900 km | 11 hours | (Average Speed) |
Therefore, the remaining journey must be completed at a speed of 108 km/h.
Revision Table: Journey Calculations
| Concept | Formula | Application in this problem |
|---|---|---|
| Distance from fraction | Fraction \(\times\) Total Distance | \(\frac{2}{5} \times 900 \text{ km} = 360 \text{ km}\) |
| Time (given Distance & Speed) | \(\frac{\text{Distance}}{\text{Speed}}\) | \(\frac{360 \text{ km}}{60 \text{ km/h}} = 6 \text{ hours}\) |
| Remaining Distance | Total Distance - Covered Distance | \(900 \text{ km} - 360 \text{ km} = 540 \text{ km}\) |
| Remaining Time | Total Time - Time Taken | \(11 \text{ hours} - 6 \text{ hours} = 5 \text{ hours}\) |
| Speed (given Distance & Time) | \(\frac{\text{Distance}}{\text{Time}}\) | \(\frac{540 \text{ km}}{5 \text{ hours}} = 108 \text{ km/h}\) |
Additional Information: Speed, Distance, and Time Concepts
The relationship between speed, distance, and time is fundamental in physics and mathematics problems involving motion. The basic formula is:
Speed = \(\frac{\text{Distance}}{\text{Time}}\)
From this, we can derive the other two relationships:
- Distance = Speed \(\times\) Time
- Time = \(\frac{\text{Distance}}{\text{Speed}}\)
These formulas are applicable when speed is constant. In problems like this one, where speed changes during the journey, we break the journey into segments where the speed is constant and apply these formulas to each segment. The total distance is the sum of distances of all segments, and the total time is the sum of times taken for all segments.
It's important to ensure that units are consistent (e.g., distance in km, time in hours, speed in km/h).
Important Questions from Speed Time and Distance
A car starts from point A towards point B, travelling at the speed of 20 km/h. 1 \(\frac{1}{2}\) hours later, another car starts from point A and travelling at the speed of 30 km/h and reaches 2 \(\frac{1}{2}\) hours before the first car. Find the distance between A and B.
A bus covered a distance of 162 km. If speed of this bus is 15 m/s, then what will be the time taken ?
An athlete runs an 800 m race in 96 seconds. His speed (in km / h) is:
A person has to cover a distance of 150 km in 15 hours. If he traveled with the speed of 11.8 km/hr for 10 hours. At what speed he has to travel to cover the remaining distance in the remaining time?
A and B start moving towards each other from places X and Y, respectively, at the same time on the same day. The speed of A is 20% more than that of B. After meeting on the way, A and B take p hours and \(7\frac{1}{5}\) hours, respectively, to reach Y and X, respectively. What is the value of p?
