Escape Velocity. Calculate the escape velocity from each of the following. a. The surface of Mars (mass = 0.11MEarth, radius = 0.53REarth) b. The surface of Mars’s moon Phobos (mass = 1.1 * 1016 kg, radius = 12 km) c. The cloud tops of Jupiter (mass =317.8MEarth, radius = 11.2REarth) d. Our solar system, starting from Earth’s orbit (Hint: Most of the mass of our solar system is in the Sun; MSun = 2.0 * 1030 kg.) e. Our solar system, starting from Saturn’s orbit
The Correct Answer and Explanation is :
To calculate the escape velocity for each of these objects, we use the formula:
[
v_e = \sqrt{\frac{2GM}{r}}
]
Where:
- (v_e) is the escape velocity
- (G) is the gravitational constant, (6.674 \times 10^{-11} \, \text{m}^3/\text{kg}\,\text{s}^2)
- (M) is the mass of the object (in kg)
- (r) is the radius of the object (in meters)
Let’s go through each of the cases:
a. The surface of Mars
- Mass of Mars = (0.11 \, M_{\text{Earth}})
- Radius of Mars = (0.53 \, R_{\text{Earth}})
- Mass of Earth = (5.972 \times 10^{24} \, \text{kg})
- Radius of Earth = (6.371 \times 10^6 \, \text{m})
The mass of Mars:
[
M_{\text{Mars}} = 0.11 \times 5.972 \times 10^{24} \, \text{kg} = 6.569 \times 10^{23} \, \text{kg}
]
The radius of Mars:
[
r_{\text{Mars}} = 0.53 \times 6.371 \times 10^6 \, \text{m} = 3.382 \times 10^6 \, \text{m}
]
Now we calculate the escape velocity from Mars:
[
v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 6.569 \times 10^{23}}{3.382 \times 10^6}} \approx 5.03 \, \text{km/s}
]
b. The surface of Phobos (Mars’s moon)
- Mass of Phobos = (1.1 \times 10^{16} \, \text{kg})
- Radius of Phobos = 12 km = (12,000 \, \text{m})
Escape velocity for Phobos:
[
v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 1.1 \times 10^{16}}{12000}} \approx 0.26 \, \text{m/s}
]
c. The cloud tops of Jupiter
- Mass of Jupiter = (317.8 \, M_{\text{Earth}})
- Radius of Jupiter = (11.2 \, R_{\text{Earth}})
The mass of Jupiter:
[
M_{\text{Jupiter}} = 317.8 \times 5.972 \times 10^{24} \, \text{kg} = 1.897 \times 10^{27} \, \text{kg}
]
The radius of Jupiter:
[
r_{\text{Jupiter}} = 11.2 \times 6.371 \times 10^6 \, \text{m} = 7.14 \times 10^7 \, \text{m}
]
Now we calculate the escape velocity from Jupiter:
[
v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 1.897 \times 10^{27}}{7.14 \times 10^7}} \approx 59.5 \, \text{km/s}
]
d. Our solar system, starting from Earth’s orbit
- Mass of the Sun = (2.0 \times 10^{30} \, \text{kg})
- Distance from Earth to the Sun = (1.496 \times 10^{11} \, \text{m})
Escape velocity from Earth’s orbit:
[
v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 2.0 \times 10^{30}}{1.496 \times 10^{11}}} \approx 42.1 \, \text{km/s}
]
e. Our solar system, starting from Saturn’s orbit
- Distance from Earth to Saturn = (1.429 \times 10^{12} \, \text{m})
Escape velocity from Saturn’s orbit:
[
v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 2.0 \times 10^{30}}{1.429 \times 10^{12}}} \approx 12.1 \, \text{km/s}
]
Summary of Escape Velocities:
- Mars: (5.03 \, \text{km/s})
- Phobos: (0.26 \, \text{m/s})
- Jupiter: (59.5 \, \text{km/s})
- Earth’s orbit: (42.1 \, \text{km/s})
- Saturn’s orbit: (12.1 \, \text{km/s})
These velocities represent the speed at which an object must travel to escape the gravitational influence of these bodies. The escape velocity increases with mass and decreases with distance from the center of the object.
Now I’ll generate an image to visualize these velocities!
Here’s the infographic showing the escape velocities for Mars, Phobos, Jupiter, Earth’s orbit, and Saturn’s orbit. You can see how each object’s escape velocity varies based on its mass and distance from the center.
