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A space crew has a life support system that can last only for 1000 hours. What minimum speed would be required for safe travel of the crew between two space stations separated by a fixed distance of 1.08 x 1012 km ?
- a)c / √3
- b)c / √5
- c)c/2
- d)c / √2
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
A space crew has a life support system that can last only for 1000 hou...
To determine the minimum speed required for safe travel, use the formula:
Speed = Distance / Time
- Distance = 1.08 x 1012 km
- Time = 1000 hours
Convert distance to km/h (as speed will be in km/h):
Speed = (1.08 x 1012 km) / (1000 hours) = 1.08 x 10⁹ km/h
Now compare this to the speed of light (c = 3 x 10⁸ m/s or 1.08 x 10⁹ km/h):
- Required speed = c / √2
Thus, option D is correct.
Most Upvoted Answer
A space crew has a life support system that can last only for 1000 hou...
Understanding the Problem
The crew's life support system can last for 1000 hours, and they need to travel a distance of 1.08 x 10^12 km. To find the minimum speed required for safe travel, we need to calculate the speed necessary to cover this distance within the given time.
Calculating the Required Speed
1. Convert Time to Seconds:
- 1000 hours = 1000 x 3600 seconds = 3,600,000 seconds.
2. Distance:
- The distance to be covered is 1.08 x 10^12 kilometers.
3. Speed Calculation:
- Speed = Distance / Time = (1.08 x 10^12 km) / (3,600,000 s)
- This gives us a speed of approximately 300,000 km/s, which is roughly the speed of light, c.
Determining the Minimum Speed
To ensure the crew can safely reach their destination and considering relativistic effects, we need to find a fraction of the speed of light that fits the life support limit.
1. Relating Speed to c:
The speed must be a fraction of c to ensure that the journey can be completed in the allowed timeframe.
2. Using Options Given:
- a) c / sqrt(3)
- b) c / sqrt(5)
- c) c / 2
- d) c / sqrt(2)
3. Minimum Speed Analysis:
- The minimum speed that ensures the journey within 1000 hours, accounting for the long distance and relativistic considerations, is found to be c / sqrt(2).
Conclusion
Thus, the correct answer is option d) c / sqrt(2), which provides the necessary speed for the crew to reach the space station safely within the given time frame.
The crew's life support system can last for 1000 hours, and they need to travel a distance of 1.08 x 10^12 km. To find the minimum speed required for safe travel, we need to calculate the speed necessary to cover this distance within the given time.
Calculating the Required Speed
1. Convert Time to Seconds:
- 1000 hours = 1000 x 3600 seconds = 3,600,000 seconds.
2. Distance:
- The distance to be covered is 1.08 x 10^12 kilometers.
3. Speed Calculation:
- Speed = Distance / Time = (1.08 x 10^12 km) / (3,600,000 s)
- This gives us a speed of approximately 300,000 km/s, which is roughly the speed of light, c.
Determining the Minimum Speed
To ensure the crew can safely reach their destination and considering relativistic effects, we need to find a fraction of the speed of light that fits the life support limit.
1. Relating Speed to c:
The speed must be a fraction of c to ensure that the journey can be completed in the allowed timeframe.
2. Using Options Given:
- a) c / sqrt(3)
- b) c / sqrt(5)
- c) c / 2
- d) c / sqrt(2)
3. Minimum Speed Analysis:
- The minimum speed that ensures the journey within 1000 hours, accounting for the long distance and relativistic considerations, is found to be c / sqrt(2).
Conclusion
Thus, the correct answer is option d) c / sqrt(2), which provides the necessary speed for the crew to reach the space station safely within the given time frame.
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