91影视

a) Distance of the Earth from the Moon鈥檚 surface,$d_{em}=3.82脳{10}^{8}m$

b) The precision of the separation is about $未d_{em}=15\mathrm{cm}$.

c) The diameter of the spot formed on the Moon,$D=3km$

Pulses with pulsed lasers emit light not in continuous mode, but in the form of optical pulses (light intensity).

The distance between the Earth and the Moon helps to get the time that may the laser take to reach the Moon's surface from the Earth's observatory. Now, using the precision value and the uncertainty condition of time-distance, the uncertainty in time can be calculated. Again use the basic formula of tangent angle to find the angular divergence of the beam.

Formulas:

The time is taken by the light to travel the given distance back and forth,

$t=\frac{2d}{c}$ 鈥�.. (1)

The uncertainty condition of distance-time as

$2d_{em}鈭�t$

$\frac{未t}{t}=\frac{未d_{e}m}{2d_{e}m}$ 鈥�.. (2)

The angular divergence of a beam,

$胃=2ta{n}^{-1}\left(\frac{Radiusofthespot}{d_{e}m}\right)$ 鈥�.. (3)

The time value that is predicted to be taken by the laser light to reach Moon from Earth and back to Earth can be calculated using equation (1) as follows:

$$\begin{gathered} t=\frac{2\left(3.82脳{10}^{8}m\right)}{3脳{10}^{8}m/s} \\ =2.55s \end{gathered}$$

Hence, the value of the time is 2.55 s.

Using the given relation of equation (2) and the given data, the value of the uncertainty in travel time as follows:

$$\begin{gathered} 未t=\frac{t未d_{em}}{2d_{em}} \\ =\frac{\left(2.55s\right)\left(0.15m\right)}{2\left(3.82脳{10}^{8}m\right)} \\ =5.0脳{10}^{-10}s \end{gathered}$$

Hence, the value of uncertainty in time is $5.0脳{10}^{-10}s$.

The radius of the spot can be given as:

$$\begin{gathered} Radius=\frac{D}{2} \\ =\frac{3\mathrm{km}}{2} \\ =1.5\mathrm{km} \end{gathered}$$

Now, using this radius value in equation (3), the angular divergence of the beam is calculated as follows:

$$\begin{gathered} 胃=2{\tan }^{-1}\left(\frac{1.5脳{10}^{3}m}{3.82脳{10}^{8}m}\right) \\ =2{\tan }^{-1}\left(0.00000392脳{10}^{-5}\right) \\ =\left(7.84脳{10}^{-6}\right)掳 \end{gathered}$$

Hence, the value of angular divergence is $\left(7.84脳{10}^{-6}\right)掳$.