What is the approximate minimum velocity, known as escape velocity, that a rocket must have to overcome Earth gravity and travel into space?

Introduction / Context:
Escape velocity is an important concept in gravitational physics and space science. It represents the minimum speed that an object must have at the surface of a massive body, such as the Earth, so that it can move infinitely far away without falling back under gravity, assuming no additional propulsion and neglecting air resistance. This question asks for the approximate numerical value of Earth escape velocity expressed in kilometres per second.

Given Data / Assumptions:

• We are discussing escape velocity from the surface of the Earth.
• Air resistance and atmospheric drag are neglected.
• The rocket is assumed to move away without further thrust after reaching escape speed.
• We use the standard formula for escape velocity from a spherical mass.

Concept / Approach:
Escape velocity v_esc from the surface of a spherical body of mass M and radius R is given by v_esc = sqrt(2 * G * M / R), where G is the universal gravitational constant. For Earth, substituting standard values of G, M and R gives a value of approximately 11.2 kilometres per second. This means any object given this speed at Earth surface and directed outward can theoretically escape Earth gravitational field without additional propulsion, if we ignore the atmosphere.

Step-by-Step Solution:
Step 1: Write the formula for escape velocity: v_esc = sqrt(2 * G * M / R). Step 2: Insert typical values for Earth: G ≈ 6.67 * 10^-11 N·m^2·kg^-2, M ≈ 5.97 * 10^24 kg, R ≈ 6.37 * 10^6 m. Step 3: Compute the quantity inside the square root: 2 * G * M / R gives a value on the order of 1.25 * 10^8 m^2·s^-2. Step 4: Take the square root to get v_esc ≈ 1.12 * 10^4 m/s. Step 5: Convert metres per second to kilometres per second by dividing by 1000, yielding about 11.2 km/s. Step 6: Compare with the options and identify 11.2 km/s as the correct approximate escape velocity.

Verification / Alternative check:
Space science references and standard physics textbooks consistently quote Earth escape velocity as about 11.2 km/s. Launch velocities for space missions are designed with this order of magnitude in mind, although real rockets use continuous propulsion and must overcome atmospheric drag, so their actual trajectory planning is more complex. The fact that all reliable sources cluster around 11.2 km/s for Earth escape velocity confirms that this is the correct option.

Why Other Options Are Wrong:
18 km/s: This value is significantly higher than the theoretical escape velocity from Earth and would represent excess speed that is not required to just escape gravity.

21 km/s: This is almost twice the correct value and does not match standard calculations or references.

35 km/s: This is far above Earth escape velocity; values like this are associated with orbital speeds around the Sun for inner planets, not escape from Earth surface.

Common Pitfalls:
Students sometimes confuse orbital speed with escape speed or mix up units, quoting values in metres per second when the question expects kilometres per second. Another mistake is to think escape velocity depends on mass of the rocket itself, whereas it actually depends only on the mass and radius of the planet and the gravitational constant. Remember that for Earth, a good number to keep in mind is approximately 11.2 km/s.

Final Answer:
The minimum velocity required for a rocket to escape Earth gravity is approximately 11.2 km/s.