Sure, I'll provide a step-by-step solution for these calculations:
### Step-by-Step Solution:
1. Gravitational Acceleration ([tex]\( g \)[/tex]) at Height [tex]\( h \)[/tex]:
[tex]\[
g = \frac{G \cdot M}{(R + h)^2}
\][/tex]
Here:
- [tex]\( G = 6.67 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex], the gravitational constant.
- [tex]\( M = 5.972 \times 10^{24} \, \text{kg} \)[/tex], the mass of the Earth.
- [tex]\( R = 6.371 \times 10^6 \, \text{m} \)[/tex], the radius of the Earth.
- [tex]\( h = 10,000 \, \text{m} \)[/tex], the height above sea level.
Plugging in the values, we get:
[tex]\[
g = \frac{(6.67 \times 10^{-11}) \cdot (5.972 \times 10^{24})}{(6.371 \times 10^6 + 10,000)^2}
\][/tex]
Calculating this gives:
[tex]\[
g = 9.782911927479423 \, \text{m/s}^2
\][/tex]
2. Air Density ([tex]\( p \)[/tex]) at Height [tex]\( h \)[/tex]:
[tex]\[
p = p_0 \cdot \exp\left(\frac{-g \cdot \text{mw} \cdot h}{R \cdot T}\right)
\][/tex]
Here:
- [tex]\( p_0 = 1.225 \, \text{kg/m}^3 \)[/tex], the initial air density at sea level.
- [tex]\( \text{mw} = 0.0289644 \, \text{kg/mol} \)[/tex], the molecular weight of air.
- [tex]\( R = 8.3144598 \, \text{J/(mol·K)} \)[/tex], the gas constant.
- [tex]\( T = 302.594 \, \text{K} \)[/tex], the temperature in Kelvin.
Plugging in the values, we get:
[tex]\[
p = 1.225 \cdot \exp\left(\frac{-9.782911927479423 \cdot 0.0289644 \cdot 10,000}{8.3144598 \cdot 302.594}\right)
\][/tex]
Calculating this gives:
[tex]\[
p = 0.3971988269923185 \, \text{kg/m}^3
\][/tex]
3. Final Velocity ([tex]\( v_r \)[/tex]):
[tex]\[
v_r = 0 + (4600 \cdot 1609) \ln\left(\frac{m_0}{m}\right) - g
\][/tex]
Here:
- [tex]\( \ln \)[/tex] represents the natural logarithm.
- [tex]\( m_0 = 2000 \, \text{kg} \)[/tex], the initial mass of the rocket.
- [tex]\( m = 1500 \, \text{kg} \)[/tex], the final mass of the rocket.
Plugging in the values, we get:
[tex]\[
v_r = 0 + 4600 \cdot 1609 \cdot \ln\left(\frac{2000}{1500}\right) - 9.782911927479423
\][/tex]
Calculating this gives:
[tex]\[
v_r = 2129240.3081326834 \, \text{m/s}
\][/tex]
4. Drag Force ([tex]\( f_d \)[/tex]):
[tex]\[
f_d = \frac{1}{2} \cdot p \cdot v_r^2 \cdot \frac{1}{2} \cdot \ln\left(\frac{m_0}{m'}\right)
\][/tex]
Given that [tex]\( v_r \)[/tex] is initially zero, this makes:
[tex]\[
f_d = \frac{1}{2} \cdot 0.3971988269923185 \cdot 0^2 \cdot \frac{1}{2} \cdot \ln\left(\frac{2000}{1500}\right)
\][/tex]
Hence:
[tex]\[
f_d = 0.0 \, \text{N}
\][/tex]
5. Force ([tex]\( f \)[/tex]):
[tex]\[
f = \frac{\ln\left(\frac{m_0}{m}\right) \cdot v_r}{0.23305928}
\][/tex]
Plugging in the values, we get:
[tex]\[
f = \frac{\ln\left(\frac{2000}{1500}\right) \cdot 2129240.3081326834}{0.23305928}
\][/tex]
Calculating this gives:
[tex]\[
f = 2628276.65386883 \, \text{N}
\][/tex]
6. Acceleration ([tex]\( a \)[/tex]):
[tex]\[
a = \frac{f}{\ln\left(\frac{m_e}{m}\right)}
\][/tex]
Here:
- [tex]\( m_e = 1800 \, \text{kg} \)[/tex], the effective mass.
Plugging in the values, we get:
[tex]\[
a = \frac{2628276.65386883}{\ln\left(\frac{1800}{1500}\right)}
\][/tex]
Calculating this gives:
[tex]\[
a = 14415611.077954432 \, \text{m/s}^2
\][/tex]
7. Volume ([tex]\( V \)[/tex]):
[tex]\[
V = a \cdot h
\][/tex]
Plugging in the values, we get:
[tex]\[
V = 14415611.077954432 \cdot 10,000
\][/tex]
Calculating this gives:
[tex]\[
V = 144156110779.5443 \, \text{m}^3
\][/tex]
Therefore, the calculated values are:
- Gravitational acceleration, [tex]\( g = 9.782911927479423 \, \text{m/s}^2 \)[/tex]
- Air density, [tex]\( p = 0.3971988269923185 \, \text{kg/m}^3 \)[/tex]
- Drag force, [tex]\( f_d = 0.0 \, \text{N} \)[/tex]
- Final velocity, [tex]\( v_r = 2129240.3081326834 \, \text{m/s} \)[/tex]
- Force, [tex]\( f = 2628276.65386883 \, \text{N} \)[/tex]
- Acceleration, [tex]\( a = 14415611.077954432 \, \text{m/s}^2 \)[/tex]
- Volume, [tex]\( V = 144156110779.5443 \, \text{m}^3 \)[/tex]
