If you were to walk brieftly in space without a spacesuit while far from the Sun ( as an astronaut does in the movie 2001, A Space Odyssey ), you would feel the cold of space- while you radiated energy, you would absorb almost non from your environment . (a) At what rate would you lose energy ? (b) How much energy would you lose in 44s ? Assume that your emissivity is 0.90, and estimate other data needed in the calculations.

Answer

To solve the problem of energy loss in space, we will follow these steps:

### Step 1: Understand the formula for power radiated
The power radiated by an object can be calculated using the Stefan-Boltzmann law:
\[ P = \sigma \epsilon A T^4 \]
where:
- \( P \) is the power radiated (in Watts),
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.6704 \times 10^{-8} \, \text{W/m}^2 \cdot \text{K}^4 \)),
- \( \epsilon \) is the emissivity of the surface (given as 0.90),
- \( A \) is the surface area (in m²),
- \( T \) is the absolute temperature in Kelvin (K).

### Step 2: Estimate the parameters
1. **Emissivity (\( \epsilon \))**: Given as 0.90.
2. **Surface Area (\( A \))**: Average surface area of a human body is approximately \( 1.7 \, \text{m}^2 \).
3. **Body Temperature**: The average body temperature is \( 37^\circ \text{C} \). To convert this to Kelvin:
\[ T = 37 + 273 = 310 \, \text{K} \]

### Step 3: Calculate the power radiated
Now we can substitute the values into the formula:
\[ P = (5.6704 \times 10^{-8}) \times (0.90) \times (1.7) \times (310)^4 \]

Calculating \( (310)^4 \):
\[ 310^4 = 9.2361 \times 10^9 \, \text{K}^4 \]

Now substituting this back into the power formula:
\[ P = (5.6704 \times 10^{-8}) \times (0.90) \times (1.7) \times (9.2361 \times 10^9) \]

Calculating this step-by-step:
1. Calculate \( 0.90 \times 1.7 = 1.53 \).
2. Now calculate \( 5.6704 \times 10^{-8} \times 1.53 = 8.688 \times 10^{-8} \).
3. Finally, multiply by \( 9.2361 \times 10^9 \):
\[ P \approx 800 \, \text{W} \]

### Step 4: Calculate energy lost in 44 seconds
Energy lost can be calculated using the formula:
\[ E = P \times t \]
where \( E \) is the energy (in Joules), \( P \) is the power (in Watts), and \( t \) is the time (in seconds).

Substituting the values:
\[ E = 800 \, \text{W} \times 44 \, \text{s} = 35200 \, \text{J} \]

### Final Result
The energy lost in 44 seconds is approximately:
\[ E \approx 3.52 \times 10^4 \, \text{J} \text{ or } 35,200 \, \text{J} \]