Lunar astronauts placed a reflector on the Moon’s surface, off which a laser beam is periodically reflected. The distance to the Moon is calculated from the round trip time. The taken for the laser pulse to travel there and back again can be measured to 0.1 ns. What percent accuracy is this, given the average distance to the Moon is 3.84 X 108 m?
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To calculate the percentage accuracy, we need to compare the precision of the measurement (0.1 ns) to the distance to the Moon (3.84 x 10^8 m).
First, let's convert the precision of the measurement from nanoseconds (ns) to meters (m). The speed of light is approximately 3 x 10^8 m/s. Therefore, the precision of 0.1 ns corresponds to:
0.1 ns x (3 x 10^8 m/s) = 3 x 10^-2 m
Now we can find the percentage accuracy by dividing the precision of the measurement (3 x 10^-2 m) by the average distance to the Moon (3.84 x 10^8 m) and multiplying by 100:
(3 x 10^-2 m / 3.84 x 10^8 m) x 100 = (3 x 10^-2 / 3.84 x 10^8) x 100
To simplify this further, we can divide the numerator and the denominator by 3:
(1 x 10^-2 / 1.28 x 10^8) x 100
Now divide the numerator by 10^2 and the denominator by 10^8:
(10^-4 / 1.28) x 100
Finally, calculate the percentage:
(0.78125 x 10^-4) x 100 ≈ 0.78125 x 10^-2 ≈ 0.0078125%
Therefore, the percent accuracy of this measurement is approximately 0.0078125%.