Escape Velocity Calculator

Mass:

Radius:

Escape Velocity (m/s): Escape Velocity (km/s): Escape Velocity (mph): Escape Velocity (Hz): First Cosmic Velocity (m/s): First Cosmic Velocity (km/s): First Cosmic Velocity (mph):

1. What is an Escape Velocity Calculator?

Definition: This calculator computes the escape velocity (\(v_e\)), the minimum speed needed to escape a celestial body’s gravity without further acceleration.

Purpose: It is used in astrophysics and space exploration to determine the speed required for spacecraft to leave planets, moons, or stars.

2. How Does the Calculator Work?

The calculator uses these formulas:

\[ v_e = \sqrt{\frac{2GM}{r}} \] \[ v_1 = \frac{v_e}{\sqrt{2}} \]

Explanation: Input mass and radius in your chosen units. The calculator converts to base units (kg, m) and outputs escape velocity and first cosmic velocity in m/s, km/s, mph, and Hz.

Unit Conversions:

1 t = 1000 kg, 1 oz = 0.0283495 kg, 1 lb = 0.453592 kg, 1 st = 6.35029 kg
1 US ton = 907.185 kg, 1 long ton = 1016.05 kg
1 Earth = \(5.972 \times 10^{24}\) kg, 1 Sun = \(1.989 \times 10^{30}\) kg
1 km = 1000 m, 1 yd = 0.9144 m, 1 mi = 1609.34 m
1 R⊕ = 6,371,000 m, 1 R☉ = 695,000,000 m
1 m/s = 0.001 km/s = 2.23694 mph, 1 m/s = \(\frac{1}{2\pi}\) Hz

3. Importance of Escape Velocity

Details: Escape velocity is critical for space travel and understanding gravitational fields. Examples include:

Rockets need this speed to leave Earth (11.2 km/s).
It determines if probes escape planetary orbits.
It aids in studying exoplanetary systems.

Applications: Essential for mission planning, satellite launches, and astrophysical modeling.

4. Using the Calculator

Tips: Enter positive values with up to 4 decimal places and select units. Results are in m/s, km/s, mph, and Hz (escape) plus m/s, km/s, mph (first cosmic). Values < 0.0001 use scientific notation. Avoid zero radius.

Example: For Earth (\(M = 1 \, \text{Earth} = 5.972 \times 10^{24} \, \text{kg}\), \(r = 1 \, \text{R⊕} = 6371 \, \text{km}\)):

\(v_e = \sqrt{\frac{2 \times 6.674 \times 10^{-11} \times 5.972 \times 10^{24}}{6.371 \times 10^6}} \approx 11186.2330 \, \text{m/s}\)
\(v_e \approx 11.1862 \, \text{km/s}\)
\(v_e \approx 25020.6146 \, \text{mph}\)
\(v_e \approx 1780.6148 \, \text{Hz}\)
\(v_1 \approx 7909.0770 \, \text{m/s}\)
\(v_1 \approx 7.9091 \, \text{km/s}\)
\(v_1 \approx 17694.6148 \, \text{mph}\)

5. Related Concepts

First Cosmic Velocity: The speed for a circular orbit, \(v_1 = \frac{v_e}{\sqrt{2}}\), used by satellites.

Gravitational Potential: Energy per unit mass, \(U = -\frac{GM}{r}\), relates to escape energy.

Orbital Mechanics: Escape velocity exceeds orbital speeds, enabling interplanetary travel.

6. Frequently Asked Questions (FAQ)

Q: What’s the difference between escape and first cosmic velocity?
A: Escape velocity (\(v_e\)) is for leaving gravity entirely; first cosmic velocity (\(v_1\)) is for orbiting.

Q: Can mass or radius be negative?
A: No, both must be positive for physical meaning.

Q: Why does the result show zero?
A: If radius is zero, division by zero occurs, so results default to zero.

Q: Why are some results in scientific notation?
A: Values < 0.0001 are displayed as, e.g., \(1.23 \times 10^{-5}\), for clarity.

Q: How do I use Earth or Sun units?
A: Enter the value (e.g., 1), select "Earths" or "Suns" for mass, or "R⊕" or "R☉" for radius, and it scales accordingly.