Calculate the escape velocity of the moon . Given accelaration due to gravity on the moon =1.62 m/s^2 and radius of the moon =1.738*10^6 m
The Correct Answer and Explanation is:
Answer:
To calculate the escape velocity of the Moon, we use the formula: ve=2gRv_e = \sqrt{2 g R}ve=2gR
Where:
- vev_eve is the escape velocity,
- ggg is the acceleration due to gravity,
- RRR is the radius of the Moon.
Given:
- g=1.62 m/s2g = 1.62 \, \text{m/s}^2g=1.62m/s2
- R=1.738×106 mR = 1.738 \times 10^6 \, \text{m}R=1.738×106m
Step-by-step calculation:
ve=2×1.62×1.738×106v_e = \sqrt{2 \times 1.62 \times 1.738 \times 10^6}ve=2×1.62×1.738×106 ve=5.62728×106v_e = \sqrt{5.62728 \times 10^6}ve=5.62728×106 ve≈2372 m/sv_e \approx 2372 \, \text{m/s}ve≈2372m/s
Final Answer:
ve≈2372 m/s\boxed{v_e \approx 2372 \, \text{m/s}}ve≈2372m/s
Textbook-Style Explanation
The escape velocity of a celestial body is the minimum speed an object must attain to break free from the body’s gravitational influence without further propulsion. It depends on two key factors: the gravitational field strength (acceleration due to gravity) and the radius of the body.
On the Moon, the gravitational pull is significantly weaker than that of Earth due to its smaller mass and size. This means objects on the Moon require much less energy to escape its gravity.
The formula used to calculate escape velocity is: ve=2gRv_e = \sqrt{2 g R}ve=2gR
This equation is derived from the conservation of energy principle, where the kinetic energy of the object must be equal to the gravitational potential energy needed to escape.
Given the Moon’s surface gravity g=1.62 m/s2g = 1.62 \, \text{m/s}^2g=1.62m/s2 and its radius R=1.738×106 mR = 1.738 \times 10^6 \, \text{m}R=1.738×106m, we substitute these into the formula: ve=2×1.62×1.738×106=5.62728×106≈2372 m/sv_e = \sqrt{2 \times 1.62 \times 1.738 \times 10^6} = \sqrt{5.62728 \times 10^6} \approx 2372 \, \text{m/s}ve=2×1.62×1.738×106=5.62728×106≈2372m/s
This result tells us that an object must be launched from the Moon’s surface at a speed of approximately 2372 meters per second (or about 2.37 km/s) to overcome lunar gravity and escape into space.
For comparison, the escape velocity on Earth is around 11.2 km/s, which is much higher due to Earth’s stronger gravitational field. This difference is one reason why space missions launched from the Moon would require less fuel and energy than from Earth.
