Understanding the Meeting Point: Distance, Speed, and Time Calculation
This problem involves calculating the distance from a starting point (Place P) where two individuals, X and Y, will meet. They start at different times and travel along the same path at different speeds. We need to find the distance from Place P where the faster person (Y) catches up to the slower person (X).
Key Information Extraction
- Person X starts at 7:00 a.m.
- Person Y starts at 7:10 a.m.
- Speed of X (\(v_X\)): 4.8 km/hr
- Speed of Y (\(v_Y\)): 6 km/hr
- Starting Point: Place P
- Path: Same for both X and Y
Step-by-Step Solution
1. Calculate the Head Start of Person X
Person X starts 10 minutes earlier than Person Y. We need to find out how far X has traveled in these 10 minutes.
- Time difference (\(\Delta t\)): 10 minutes
- Convert time difference to hours: \(\Delta t = \frac{10}{60} \text{ hours} = \frac{1}{6} \text{ hours}\)
- Distance covered by X in this time (head start distance, \(d_{headstart}\)): \(d_{headstart} = \text{Speed of X} \times \Delta t\) \(d_{headstart} = 4.8 \text{ km/hr} \times \frac{1}{6} \text{ hours}\) \(d_{headstart} = \frac{4.8}{6} \text{ km} = 0.8 \text{ km}\)
So, when Y starts at 7:10 a.m., X is already 0.8 km ahead from Place P.
2. Calculate the Relative Speed
Since Y is faster than X and is trying to catch up, we calculate the relative speed at which Y gains on X.
- Relative Speed (\(v_{relative}\)): Speed of Y - Speed of X \(v_{relative} = v_Y - v_X\) \(v_{relative} = 6 \text{ km/hr} - 4.8 \text{ km/hr}\) \(v_{relative} = 1.2 \text{ km/hr}\)
This means Y closes the gap between them by 1.2 km every hour.
3. Calculate the Time Taken for Y to Catch Up
Now we find how long it takes for Y to cover the 0.8 km head start that X had.
- Time to catch up (\(t_{catchup}\)): \(\frac{\text{Head Start Distance}}{\text{Relative Speed}}\) \(t_{catchup} = \frac{d_{headstart}}{v_{relative}}\) \(t_{catchup} = \frac{0.8 \text{ km}}{1.2 \text{ km/hr}}\) \(t_{catchup} = \frac{8}{12} \text{ hours} = \frac{2}{3} \text{ hours}\)
It will take Y \(\frac{2}{3}\) hours (or 40 minutes) to catch up with X after Y starts.
4. Calculate the Distance from Place P where X Meets Y
We can now calculate the distance from Place P where they meet. We can use Y's speed and the time Y traveled to catch up.
- Meeting Distance (\(d_{meet}\)): Speed of Y \(\times\) Time to Catch Up \(d_{meet} = v_Y \times t_{catchup}\) \(d_{meet} = 6 \text{ km/hr} \times \frac{2}{3} \text{ hours}\) \(d_{meet} = \frac{12}{3} \text{ km}\) \(d_{meet} = 4 \text{ km}\)
Verification (Optional)
Let's verify this using X's journey. X travels for 10 minutes (1/6 hour) longer than Y. The total time X travels until they meet is:
- Total time for X (\(t_X\)): \(\Delta t + t_{catchup}\) \(t_X = \frac{1}{6} \text{ hours} + \frac{2}{3} \text{ hours}\) \(t_X = \frac{1}{6} \text{ hours} + \frac{4}{6} \text{ hours} = \frac{5}{6} \text{ hours}\)
- Distance covered by X: Speed of X \(\times\) Total time for X Distance = \(4.8 \text{ km/hr} \times \frac{5}{6} \text{ hours}\) Distance = \(\frac{4.8 \times 5}{6} \text{ km} = \frac{24}{6} \text{ km} = 4 \text{ km}\)
Both calculations confirm that X and Y meet 4 km away from Place P.
Conclusion
X and Y will meet at a distance of 4 km from Place P.
Similar Questions
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Important Questions from Speed Time and Distance
A train travelling at a speed of 72 km/hr crosses a post in 20 seconds. If it crosses another train travelling at a speed of 54 km/hr in the same direction in 1 minute 45 seconds, then the difference in length between the two trains is
Rajiv's boat can travel along the current at the 8 km/hour and against the current at the rate 6 km/hour. Find the time taken by the boat to sail 28 km in still water.
Rohit and Dinesh are 64 km apart. Rohit can walk at a speed of 15 km/hr and Dinesh at the speed of 17 km/hr. In how many hours will they meet if they are travelling towards each other?
Two trains running in opposite directions cross a man standing on the platform in 25 seconds and 32 seconds respectively and they cross each other in 30 seconds. The ratio of their speed is:
A worker covers a distance of 81 km in 11 hours. He travels partly on foot at 4.5 km/h and partly on bicycle at 15 km/h. What is the distance covered on the cycle?
