A glow-worm of mass `5.0 g` emits red light `(650 nm)` with a power of `0.10 w`. Entirely in the backward direction. To what speed will it have accelerated sfter `10 y` if released into free space and assumed to live?

Options

A. `21 ms^(-1)`
B. `29 ms^(-1)`
C. `31.8 ms^(-1)`
D. `0.08 ms^(-1)`

Explanation

To solve the problem step by step, we will follow the calculations outlined in the video transcript while providing clear explanations for each step.

### Step 1: Write down the given data
- Mass of the glow-worm, \( m = 5.0 \, \text{g} = 5.0 \times 10^{-3} \, \text{kg} \)
- Wavelength of emitted light, \( \lambda = 650 \, \text{nm} = 650 \times 10^{-9} \, \text{m} \)
- Power of emitted light, \( P = 0.10 \, \text{W} \)
- Time, \( t = 10 \, \text{years} = 10 \times 365 \times 24 \times 60 \times 60 \, \text{s} = 10 \times 3.156 \times 10^7 \, \text{s} \)

### Step 2: Calculate the total time in seconds
\[ t = 10 \times 3.156 \times 10^7 \, \text{s} = 3.156 \times 10^8 \, \text{s} \]

### Step 3: Relate power to energy and momentum
The power \( P \) can be related to the energy emitted per unit time. The energy of a single photon is given by:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \, \text{J s} \)) and \( c \) is the speed of light (\( 3.0 \times 10^8 \, \text{m/s} \)).

### Step 4: Calculate the number of photons emitted in time \( t \)
The total energy emitted in time \( t \) is:
\[ E_{\text{total}} = P \cdot t \]
The number of photons \( n \) emitted is given by:
\[ n = \frac{E_{\text{total}}}{E} = \frac{P \cdot t}{E} = \frac{P \cdot t \cdot \lambda}{hc} \]

### Step 5: Substitute the values into the equation
Substituting the values we have:
\[ n = \frac{(0.10 \, \text{W}) \cdot (3.156 \times 10^8 \, \text{s}) \cdot (650 \times 10^{-9} \, \text{m})}{(6.626 \times 10^{-34} \, \text{J s}) \cdot (3.0 \times 10^8 \, \text{m/s})} \]

### Step 6: Calculate the momentum
The momentum \( p \) of the emitted photons is given by:
\[ p = n \cdot \frac{h}{\lambda} \]
Using the relation \( mv = p \), we can express the speed \( v \) as:
\[ mv = n \cdot \frac{h}{\lambda} \]
Thus,
\[ v = \frac{n \cdot h}{m \cdot \lambda} \]

### Step 7: Substitute \( n \) back into the equation for \( v \)
Substituting \( n \) into the equation for \( v \):
\[ v = \frac{P \cdot t \cdot h}{m \cdot c \cdot \lambda} \]

### Step 8: Substitute all known values
Substituting the known values:
\[ v = \frac{(0.10) \cdot (3.156 \times 10^8) \cdot (6.626 \times 10^{-34})}{(5.0 \times 10^{-3}) \cdot (3.0 \times 10^8) \cdot (650 \times 10^{-9})} \]

### Step 9: Calculate the speed \( v \)
Calculating the above expression gives:
\[ v \approx 21 \, \text{m/s} \]

### Final Answer
The speed to which the glow-worm will have accelerated after 10 years is approximately \( 21 \, \text{m/s} \).