Notice that
m has canceled out of the equation. The escape velocity is the same for all objects, regardless of mass. Also, we are not restricted to the surface of the planet;
R can be any starting point beyond the surface of the planet.
Escape from earth
What is the escape speed from the surface of Earth? Assume there is no energy loss from air resistance. Compare this to the escape speed from the Sun, starting from Earth’s orbit.
Strategy
We use
[link] , clearly defining the values of
R and
M . To escape Earth, we need the mass and radius of Earth. For escaping the Sun, we need the mass of the Sun, and the orbital distance between Earth and the Sun.
Solution
Substituting the values for Earth’s mass and radius directly into
[link] , we obtain
That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use
and
. The result is
or about 42 km/s.
Significance
The speed needed to escape the Sun (leave the solar system) is nearly four times the escape speed from Earth’s surface. But there is help in both cases. Earth is rotating, at a speed of nearly 1.7 km/s at the equator, and we can use that velocity to help escape, or to achieve orbit. For this reason, many commercial space companies maintain launch facilities near the equator. To escape the Sun, there is even more help. Earth revolves about the Sun at a speed of approximately 30 km/s. By launching in the direction that Earth is moving, we need only an additional 12 km/s. The use of gravitational assist from other planets, essentially a gravity slingshot technique, allows space probes to reach even greater speeds. In this slingshot technique, the vehicle approaches the planet and is accelerated by the planet’s gravitational attraction. It has its greatest speed at the closest point of approach, although it decelerates in equal measure as it moves away. But relative to the planet, the vehicle’s speed far before the approach, and long after, are the same. If the directions are chosen correctly, that can result in a significant increase (or decrease if needed) in the vehicle’s speed relative to the rest of the solar system.
Visit this
website to learn more about escape velocity.
Check Your Understanding If we send a probe out of the solar system starting from Earth’s surface, do we only have to escape the Sun?
The probe must overcome both the gravitational pull of Earth and the Sun. In the second calculation of our example, we found the speed necessary to escape the Sun from a distance of Earth’s orbit, not from Earth itself. The proper way to find this value is to start with the energy equation,
[link] , in which you would include a potential energy term for both Earth and the Sun.
As stated previously, escape velocity can be defined as the initial velocity of an object that can escape the surface of a moon or planet. More generally, it is the speed at
any position such that the
total energy is zero. If the total energy is zero or greater, the object escapes. If the total energy is negative, the object cannot escape. Let’s see why that is the case.
