Respuesta :
Answer:
To solve these problems, we can use the principles of conservation of energy. The total mechanical energy of the object at any point in its motion (kinetic energy + gravitational potential energy) remains constant.
Let's denote:
- \( v_i \) as the initial speed of the object
- \( v_f \) as the final speed of the object when it's far from Saturn
- \( R \) as the radius of Saturn (60300 km or 6.03 x 10^7 m)
- \( G \) as the gravitational constant (6.674 x 10^(-11) m^3/kg/s^2)
- \( M \) as the mass of Saturn (5.7 x 10^26 kg)
(a) To find the initial speed needed for the final speed of 24500 m/s:
Using conservation of energy, we have:
Initial kinetic energy + Initial gravitational potential energy = Final kinetic energy + Final gravitational potential energy
\( \frac{1}{2} m v_i^2 - \frac{G M m}{R} = \frac{1}{2} m v_f^2 - \frac{G M m}{\infty} \)
Where \( m \) is the mass of the object, which will cancel out from both sides of the equation.
Substituting the given values:
\( \frac{1}{2} v_i^2 - \frac{G M}{R} = \frac{1}{2} v_f^2 \)
\( \frac{1}{2} v_i^2 - \frac{G \times 5.7 \times 10^{26}}{6.03 \times 10^7} = \frac{1}{2} (24500)^2 \)
Now, solve for \( v_i \).
(b) To find the initial speed needed for the final speed of 0 m/s (escape speed):
In this case, the final kinetic energy would be 0, as the object comes to rest infinitely far away from Saturn.
Using the same conservation of energy equation:
\( \frac{1}{2} v_i^2 - \frac{G M}{R} = 0 \)
\( \frac{1}{2} v_i^2 = \frac{G M}{R} \)
Now, solve for \( v_i \).
Explanation:
Sure, let's solve both parts:
(a) For the initial speed needed so that the final speed is 24500 m/s:
\[ \frac{1}{2} v_i^2 - \frac{G M}{R} = \frac{1}{2} (24500)^2 \]
\[ \frac{1}{2} v_i^2 - \frac{6.674 \times 10^{-11} \times 5.7 \times 10^{26}}{6.03 \times 10^7} = \frac{1}{2} (24500)^2 \]
\[ \frac{1}{2} v_i^2 - 5.943 \times 10^7 = 3.00625 \times 10^8 \]
\[ \frac{1}{2} v_i^2 = 3.00625 \times 10^8 + 5.943 \times 10^7 \]
\[ \frac{1}{2} v_i^2 = 3.600625 \times 10^8 \]
\[ v_i^2 = 7.20125 \times 10^8 \]
\[ v_i = \sqrt{7.20125 \times 10^8} \]
\[ v_i \approx 26810.2 \, \text{m/s} \]
So, the initial speed needed is approximately \(26810.2 \, \text{m/s}\).
(b) For the initial speed needed so that the final speed is 0 m/s (escape speed):
\[ \frac{1}{2} v_i^2 = \frac{G M}{R} \]
\[ \frac{1}{2} v_i^2 = \frac{6.674 \times 10^{-11} \times 5.7 \times 10^{26}}{6.03 \times 10^7} \]
\[ \frac{1}{2} v_i^2 = 5.943 \times 10^7 \]
\[ v_i^2 = 1.1886 \times 10^8 \]
\[ v_i = \sqrt{1.1886 \times 10^8} \]
\[ v_i \approx 10912.1 \, \text{m/s} \]
So, the initial speed needed for the object to escape Saturn's gravity is approximately \(10912.1 \, \text{m/s}\).
