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Two artificial satellites (A and B) of earth are orbiting in equatorial plane at such altitudes that their periods of revolution are 4 hours and 16 hours respectively. Sense of rotation of satellite A is west to east while that of B is opposite. After being nearest to each other and lying in a given longitude of earth at a given point of time, the minimum time interval after which both are again nearest to each other and lying in the same longitude is x hours. Find 10x.

Question

Two artificial satellites (A and B) of earth are orbiting in equatorial plane at such altitudes that their periods of revolution are 4 hours and 16 hours respectively. Sense of rotation of satellite A is west to east while that of B is opposite. After being nearest to each other and lying in a given longitude of earth at a given point of time, the minimum time interval after which both are again nearest to each other and lying in the same longitude is x hours. Find 10x.

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Solution

This problem can be solved by understanding the relative motion of the two satellites.

  1. First, let's find the number of revolutions each satellite makes in 24 hours. Satellite A, with a period of 4 hours, makes 24/4 = 6 revolutions in 24 hours. Satellite B, with a period of 16 hours, makes 24/16 = 1.5 revolutions in 24 hours.

  2. Since Satellite A is moving from west to east and Satellite B is moving from east to west, their relative motion will be the sum of their individual motions. So, in 24 hours, they will make 6 + 1.5 = 7.5 relative revolutions.

  3. To find when they will be nearest to each other again, we need to find when they will complete a whole number of relative revolutions. This will happen when they have made 8 relative revolutions.

  4. To find how long it will take for them to make 8 relative revolutions, we divide the total time (24 hours) by the number of relative revolutions they make in that time (7.5), and then multiply by the number of relative revolutions we want (8). This gives us (24/7.5) * 8 = 25.6 hours.

  5. The problem asks for 10x, where x is the time in hours. So, 10x = 10 * 25.6 = 256.

So, the answer is 256.

This problem has been solved

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