Light of wavelength `lamda` from a small 0.5 mW He-Ne laser source, used in the school laboratory, shines from a spacecraft of mass 1000 kg. Estimate the time needed for the spacecraft to reach a velocity of `1.0km^(-1)` from rest. The momentum p of a photon of wavelength `lamda` is given by `p=(h)/(lamda)`, where h is Planck's constant.

Options

A. `6..10^(18)`
B. `3xx10^(17)`
C. `6xx10^(17)`
D. `2xx10^(15)`

Explanation

To solve the problem step by step, we can follow these calculations:

### Step 1: Calculate the Energy Emitted by the Laser
The power of the laser is given as \( P = 0.5 \, \text{mW} = 0.5 \times 10^{-3} \, \text{W} \).

### Step 2: Calculate the Number of Photons Emitted per Second
The energy of a single photon is given by the equation:
\[ E = \frac{hc}{\lambda} \]
where:
- \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{Js} \)),
- \( c \) is the speed of light (\( 3 \times 10^8 \, \text{m/s} \)),
- \( \lambda \) is the wavelength of the light.

The number of photons emitted per second, \( N \), can be calculated using the formula:
\[ N = \frac{P}{E} = \frac{P \lambda}{hc} \]

### Step 3: Calculate the Momentum of a Photon
The momentum \( p \) of a photon is given by:
\[ p = \frac{h}{\lambda} \]

### Step 4: Calculate the Rate of Change of Momentum
The total momentum imparted to the spacecraft per second (which is the force) is given by:
\[ F = Np = N \left(\frac{h}{\lambda}\right) \]
Substituting \( N \) from Step 2:
\[ F = \left(\frac{P \lambda}{hc}\right) \left(\frac{h}{\lambda}\right) = \frac{P}{c} \]

### Step 5: Relate Force to Acceleration
Using Newton's second law, we can relate the force to the acceleration \( a \):
\[ F = ma \]
where \( m = 1000 \, \text{kg} \) (mass of the spacecraft). Thus,
\[ a = \frac{F}{m} = \frac{P}{mc} \]

### Step 6: Calculate the Acceleration
Substituting the values:
\[ a = \frac{0.5 \times 10^{-3}}{1000 \times 3 \times 10^8} = \frac{0.5 \times 10^{-3}}{3 \times 10^{11}} = \frac{0.5}{3} \times 10^{-14} \, \text{m/s}^2 \]
\[ a \approx 1.67 \times 10^{-15} \, \text{m/s}^2 \]

### Step 7: Calculate the Time to Reach the Desired Velocity
Using the equation of motion:
\[ v = u + at \]
Since the spacecraft starts from rest, \( u = 0 \):
\[ v = at \implies t = \frac{v}{a} \]
Substituting \( v = 1 \, \text{km/s} = 1000 \, \text{m/s} \):
\[ t = \frac{1000}{1.67 \times 10^{-15}} \approx 5.99 \times 10^{17} \, \text{s} \]

### Final Answer
The time needed for the spacecraft to reach a velocity of \( 1.0 \, \text{km/s} \) from rest is approximately \( 5.99 \times 10^{17} \) seconds.

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