Interactive physics simulator
Escape Velocity
Escape velocity is the minimum launch speed needed to leave a planet or moon’s gravity without more propulsion. It depends on the body’s mass and the launch distance from its center.
Escape Velocity Simulator
Launch a rocket from a planet, compare its speed with escape velocity, and watch whether the path falls back, barely escapes, or escapes with extra speed.
Live Telemetry
- Planet
- Earth
- Escape velocity
- 11.19 km/s
- Launch speed
- 10.00 km/s
- Speed ratio
- 89.4%
- Altitude
- 0 km
- Live speed
- 10.00 km/s
- Specific energy
- Bound
- Outcome
- Falls back
What is Escape Velocity?
Escape velocity is the minimum speed required for an object to escape a gravitational field and reach a very large distance with zero speed left. For a planet or moon, the ideal escape velocity is found from energy conservation: initial kinetic energy must be enough to overcome the negative gravitational potential energy.
Key Definition
Escape velocity is the minimum speed needed to escape gravity without further propulsion.
- It is a speed, so it has magnitude.
- It depends on the mass of the planet or moon.
- It also depends on distance from the planet’s center.
- It does not depend on the rocket’s mass in the ideal formula.
- It is lower at higher altitude.
Escape Velocity Formula
Meaning: escape velocity equals the square root of two times gravitational constant times planet mass divided by distance from the planet’s center.
- vesc = escape velocity
- G = 6.674 × 10-11 N·m²/kg²
- M = mass of the planet, moon, or star
- r = distance from the body’s center
Escape Velocity vs Orbital Velocity
| Feature | Escape Velocity | Orbital Velocity |
|---|---|---|
| Purpose | Leave gravity | Stay in orbit |
| Formula | √(2GM/r) | √(GM/r) |
| At same radius | √2 times orbital speed | Lower speed |
| Path | Open path away | Closed orbit |
Planet Comparison
Approximate surface escape velocities:
- Moon: 2.38 km/s
- Mars: 5.03 km/s
- Earth: 11.19 km/s
- Jupiter: about 59.5 km/s
Jupiter’s escape velocity is much larger because Jupiter has far more mass than Earth.
Solved Examples
Find the escape velocity from Earth using G = 6.674 × 10-11 N·m²/kg², M = 5.972 × 1024 kg, and R = 6.371 × 106 m.
- Use the escape velocity formula: vesc = √(2GM/R).
- Substitute the values: vesc = √((2 × 6.674 × 10-11 × 5.972 × 1024) / (6.371 × 106)).
- The value inside the square root is about 1.25 × 108 m²/s².
- Taking the square root gives vesc ≈ 11,186 m/s.
- Convert to kilometers per second: 11,186 m/s ≈ 11.2 km/s.
Answer: vesc ≈ 11.2 km/s
A spacecraft is already 400 km above Earth. Estimate the escape velocity at that altitude. Use Earth radius R = 6,371 km.
- Find distance from Earth’s center: r = 6,371 km + 400 km = 6,771 km = 6.771 × 106 m.
- Use vesc = √(2GM/r).
- Substitute Earth’s GM value: GM ≈ 3.986 × 1014 m³/s².
- vesc = √((2 × 3.986 × 1014) / (6.771 × 106)).
- vesc ≈ 10,850 m/s = 10.85 km/s.
Answer: vesc ≈ 10.85 km/s
Mars has an escape velocity of about 5.03 km/s. If a rocket leaves Mars at 6.00 km/s, what is its excess speed far away from Mars?
- Use conservation of energy: v∞ = √(vlaunch2 - vesc2).
- Substitute values: v∞ = √(6.00² - 5.03²) km/s.
- Calculate: 6.00² = 36.00 and 5.03² ≈ 25.30.
- v∞ = √(10.70) ≈ 3.27 km/s.
- This positive value means the rocket escapes and still has speed left at a very large distance.
Answer: v∞ ≈ 3.27 km/s
The Moon’s escape velocity is about 2.38 km/s. A probe launches upward at 2.00 km/s. Does it escape?
- Compare launch speed with escape velocity.
- The launch speed is 2.00 km/s, which is less than 2.38 km/s.
- With no additional engine thrust and ignoring atmosphere, the probe does not have enough kinetic energy to reach infinity.
- It will rise, slow down, stop at a maximum height, and then fall back toward the Moon.
Answer: No, 2.00 km/s is less than 2.38 km/s
Common Mistakes
- Thinking escape velocity depends on rocket mass.
- Using planet radius but forgetting the altitude above the surface.
- Confusing escape velocity with orbital velocity.
- Forgetting the formula ignores air resistance and rocket thrust after launch.
- Thinking a rocket must keep escape velocity forever.
- Using diameter instead of radius in the formula.
Quick Summary
- Escape velocity is the minimum speed to leave gravity.
- Formula: vesc = √(2GM/r).
- It depends on planet mass and distance from the center.
- It does not depend on the escaping object’s mass.
- At higher altitude, escape velocity is lower.
- Launching faster than escape velocity leaves excess speed.
Energy Explanation
Escape velocity comes from setting total mechanical energy equal to zero:
The mass m of the rocket appears in both terms, so it cancels. That is why ideal escape velocity is independent of rocket mass.
Practice Questions
1. What does escape velocity mean?
Escape velocity is the minimum speed needed to leave a planet or moon’s gravity without more engine thrust, ignoring air resistance.
2. Does escape velocity depend on the mass of the rocket?
No. In the ideal formula vesc = √(2GM/r), the rocket mass cancels out. It depends on the central body’s mass M and the distance r from its center.
3. If a planet has larger mass but the same radius, does escape velocity increase?
Yes. A larger planet mass gives a stronger gravitational field, so escape velocity increases.
4. If a rocket starts at a higher altitude, does escape velocity increase or decrease?
It decreases because the rocket starts farther from the planet’s center, where gravity is weaker.
5. Earth’s escape velocity is about 11.2 km/s. Would a 9 km/s rocket escape Earth with no more thrust?
No. 9 km/s is less than 11.2 km/s, so it is below the ideal escape speed from Earth’s surface.
6. What is the relation between escape velocity and circular orbital velocity at the same radius?
vesc = √2 × vorb. Escape velocity is about 1.414 times the circular orbital velocity at the same radius.
FAQ
Frequently Asked Questions
What is escape velocity?
Escape velocity is the minimum speed an object needs to escape a planet, moon, or star’s gravitational field without further propulsion, ignoring air resistance.
What is the formula for escape velocity?
The formula is vesc = √(2GM/r), where G is the gravitational constant, M is the mass of the planet or star, and r is the distance from its center.
What is the SI unit of escape velocity?
The SI unit is meter per second (m/s). It is also commonly written in kilometers per second (km/s) for planets and spacecraft.
Does escape velocity depend on rocket mass?
No. In the ideal physics formula, the rocket mass cancels out. Escape velocity depends on the planet’s mass and the starting distance from its center.
Why is Earth’s escape velocity about 11.2 km/s?
Earth has a large mass and a radius of about 6,371 km. Substituting those values into vesc = √(2GM/R) gives about 11.2 km/s at the surface.
Is escape velocity the same as orbital velocity?
No. Orbital velocity keeps an object in orbit, while escape velocity lets it leave the gravitational field. At the same radius, escape velocity is √2 times circular orbital velocity.
Does escape velocity decrease with altitude?
Yes. As altitude increases, the distance r from the planet’s center increases, so vesc = √(2GM/r) becomes smaller.
What happens if launch speed is less than escape velocity?
If no more thrust is provided, the object is gravitationally bound. It may rise and then fall back, or it may enter an orbit if it has enough sideways velocity.
What happens if launch speed is greater than escape velocity?
The object escapes and has excess speed left far from the planet. This excess speed is called v infinity or v∞.
Does the escape velocity formula include air resistance?
No. The standard formula ignores air resistance, rocket thrust after launch, planet rotation, and atmospheric heating. Real missions need more detailed calculations.
Why does Jupiter have a much larger escape velocity than Earth?
Jupiter has much more mass than Earth. Even though its radius is also larger, its strong gravity makes the escape velocity much higher.
How does the simulator calculate escape velocity?
The simulator uses vesc = √(2GM/r), compares it with launch speed, and uses specific mechanical energy to estimate whether the rocket escapes, falls back, or leaves with excess speed.