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?
To calculate the percentage accuracy, we need to compare the measured time with the actual time it should take for the laser pulse to travel to the Moon and back.
The actual time can be calculated by dividing the distance to the Moon by the speed of light.
Speed of light = 299,792,458 meters per second (m/s)
Distance to the Moon = 3.84 X 10^8 meters
The time it should take for the laser pulse to travel to the Moon and back is:
Time = (2 * Distance) / Speed of light
Time = (2 * 3.84 X 10^8 meters) / 299,792,458 m/s
Time = 0.010881679 seconds
Now, let's calculate the percentage accuracy:
Measured time = 0.1 nanoseconds
Actual time = 0.010881679 seconds
Percentage Accuracy = ((Measured time - Actual time) / Actual time) * 100
Percentage Accuracy = ((0.1 ns - 0.010881679 seconds) / 0.010881679 seconds) * 100
After performing the calculation, the percentage accuracy is approximately 816,451%.
To calculate the percent accuracy of measuring the distance to the Moon, we need to compare the uncertainty in the measurement to the actual average distance.
First, let's convert the uncertainty in time to an uncertainty in distance. The time uncertainty given is 0.1 ns (nanoseconds), and we know that the speed of light is approximately 3 x 10^8 meters per second.
Since the laser beam travels to the Moon and back, we divide the time uncertainty by 2 to find the uncertainty in one direction.
Uncertainty in distance = (uncertainty in time) x (speed of light) / 2
Uncertainty in distance = (0.1 ns) x (3 x 10^8 m/s) / 2
Uncertainty in distance = 1.5 x 10^-5 meters
Now, we can calculate the percent accuracy by dividing the uncertainty in distance by the average distance to the Moon and multiplying by 100.
Percent accuracy = (uncertainty in distance / average distance) x 100
Percent accuracy = (1.5 x 10^-5 m / 3.84 x 10^8 m) x 100
Percent accuracy ≈ 3.9 x 10^-6 %
So, the laser pulse measurement method has an accuracy of approximately 3.9 x 10^-6 %.
.1 ns = 1 * 10^-10 m/s
c = 3*10^8 m/s approx
d = 2 * 3.84 * 10^8 = 7.68 *10^8 m
time = d/c = 7.68/3 = 2.56 s
percent = (10^-10/2.56) * 100
= 10^-8/2.56
= 10 * 10^-9 /2.56
= 3.9 * 10^-9 percent