The moon would depart from the gravitational field of earth forever, if its speed increased nearly by (in percent) :

Question

Accepted Answer

To understand why Option 3 is the correct answer, we need to consider the concept of escape velocity and how it applies to the moon's orbit around the Earth.

1. **Explanation for Option 3 (42% increase):**
   - The moon is currently in a stable orbit around the Earth due to the balance between its gravitational attraction to the Earth and its orbital velocity. To escape Earth's gravitational field, the moon would need to reach a speed that allows it to overcome this gravitational pull completely.
   - The escape velocity from Earth is approximately 11.2 km/s. The moon's current orbital speed is about 1.022 km/s. To calculate the percentage increase needed for the moon to reach escape velocity, we use the formula for percentage increase: $(\text{Escape Velocity} - \text{Current Velocity}) / \text{Current Velocity} \times 100\%$.
   - Substituting the values: $(11.2 - 1.022) / 1.022 \times 100\% \approx 996.08\%$. However, this calculation is for an object starting from the Earth's surface. For the moon, which is already in orbit, the increase needed is significantly less due to its initial velocity and distance from Earth.
   - The correct theoretical calculation for an object in orbit, like the moon, shows that an increase of approximately 42% in its current orbital speed would allow it to reach escape velocity from its orbital position. This is why Option 3 is correct.

2. **Explanation for why other options are incorrect:**
   - **Option 1 (22% increase):** A 22% increase in the moon's speed would not be sufficient to reach the escape velocity needed to leave Earth's gravitational field. This increase would not provide the necessary kinetic energy to overcome the gravitational pull.
   - **Option 2 (24% increase):** Similar to Option 1, a 24% increase is also insufficient. It falls short of the required speed to achieve escape velocity from the moon's current orbit.
   - **Option 4 (38% increase):** While closer to the correct value, a 38% increase still does not meet the threshold needed for the moon to escape Earth's gravitational influence. It is slightly below the calculated requirement.

In conclusion, Option 3, with a 42% increase in speed, is the correct answer because it accurately reflects the necessary increase in the moon's velocity to achieve escape velocity from its current orbital position around Earth.