Problem 81 - Master Resource Book in JEE Main Physics

Question

If the moon is to escape from the gravitational field of the earth forever, it will require a velocity (a) \(11.2 \mathrm{kms}^{-1}\) (b) less than \(11.2 \mathrm{kms}^{-1}\) (c) slightly more than \(11.2 \mathrm{kms}^{-1}\) (d) \(22.4 \mathrm{kms}^{-1}\)

Step-by-Step Solution

Answer

(c) slightly more than 11.2 kms^{-1}

Step 1: Understanding Escape Velocity

Escape velocity is the minimum velocity an object must have to break free from the gravitational attraction of a celestial body without further propulsion. For Earth, this velocity is calculated considering its gravitational force.

Step 2: Escape Velocity of Earth

The escape velocity from Earth's surface is approximately 11.2 km/s. This is derived from the formula \( v = \sqrt{2 g R} \), where \( v \) is the escape velocity, \( g \) is the acceleration due to gravity \( (9.81 \, \text{m/s}^2) \), and \( R \) is the radius of the Earth \( (6.371 \times 10^6 \, \text{m}) \).

Step 3: Applying to the Moon

Since the moon orbits the Earth, if it needs to escape Earth's gravity, it must achieve a velocity slightly more than the escape velocity from the Earth's surface. This ensures it overcomes any additional gravitational effects.

Step 4: Conclusion on Required Velocity

Based on the known escape velocity from Earth's surface (11.2 km/s), and considering the moon's position in Earth's gravitational field, a velocity slightly more than 11.2 km/s is required for it to escape Earth's gravitational pull completely.

Key Concepts

Gravitational Attraction

Gravitational attraction is the force that pulls objects toward one another. It's a key component of what keeps planets, moons, and other celestial bodies in stable orbits around larger masses such as stars or planets. The laws governing this force were laid out by Sir Isaac Newton and further refined by Albert Einstein's theory of General Relativity.
Newton's Law of Universal Gravitation states that every point mass attracts every other point mass in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is described by the equation:

• \[ F = G \frac{m_1 m_2}{r^2} \]
• where \( F \) is the force of attraction, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses involved, and \( r \) is the distance between the centers of the two masses.

Gravitational attraction is important because it's the reason why objects in space, like the moon, stay in orbit around Earth rather than floating off into space.

Gravity

Gravity is the force that gives weight to objects and causes them to fall toward the ground when dropped. It is the force that the Earth exerts on all objects within its vicinity. The strength of gravity on Earth is described by the acceleration due to gravity, often denoted as \( g \), which is approximately \( 9.81 \, ext{m/s}^2 \).
Gravity is not just limited to Earth. Every mass in the universe exerts some gravitational force, but it becomes notably stronger with larger masses, such as planets and stars.

• Gravity is responsible for keeping the atmosphere around the Earth and for the ocean tides created by the gravitational pull of the moon and the sun on Earth's water.
• It plays an essential role in the movement of planets around the sun and moons around planets.

In the context of escape velocity, this force is what must be overcome to send a spacecraft or any object away from the Earth or another celestial body without it falling back down.

Moon

The moon is Earth's only natural satellite and orbits the Earth at an average distance of about 384,400 kilometers. It plays a crucial role in Earth's tides due to its gravitational pull. The moon has a weaker gravity in comparison to Earth, which is about \( 1/6 \)th of Earth's gravitational pull.
When considering the escape velocity necessary for the moon to leave Earth's gravitational field, it's essential to remember that the moon is already in motion around the Earth. This means it requires only a slightly greater velocity than what's needed for an object starting from the Earth's surface to escape the planet's gravity. This is because it's already halfway "up the hill" as it orbits Earth.

• Understanding the movement of the moon and its gravitational interactions with Earth helps scientists explain its influence on ocean tides and its gradual distancing from Earth due to tidal forces.
• The moon’s gravity also affects humans and technology, influencing space explorations and potentially future human settlements.

In essence, the moon's relationship with Earth is an intricate dance governed by the fundamentals of gravity and motion.