How fast does a rocket need to be going to break free of Earth's gravity? How fast would it have to be going in order to completely leave the solar system? (For the second part of this problem, assume that the Sun is the only mass in the solar system and that the rocket is beginning its journey from Earth's orbit?)
The escape velocity for any body may be calculated from Ve = sqrt[2µ/r] where Ve = the escape velocity in feet per second, µ = the gravitational constant of the body (1.407974x10^16 ft.^3/sec.^2 for Earth) and r = the surface distance in feet. For earth, with an equatorial radius of ~3963 miles r becomes 3963(5280) = ~20,924,640 feet and Ve = sqrt[2(1.407974x10^16)/20,924,640] = ? ft./sec. Divide your answer by 1.467 to get the answer in miles per hour.
For the moon, µ = 1.731837x10^14 ft.^3/sec.^2, r = 1080 miles = 5,702, 400 feet.
For the sun, µ = 4.68772x10^21 and r = 432,495 miles = 2,283,573,600 feet.
To break free from Earth's gravity and reach orbit, a rocket typically needs to reach a speed of about 28,000 kilometers per hour (or about 17,500 miles per hour). This speed is known as the orbital velocity.
Now, to completely leave the solar system, we need to consider the escape velocity from the Sun's gravitational pull. The escape velocity is the minimum speed required for an object to escape the gravitational field of a celestial body.
To calculate the escape velocity, we can use the formula:
v = √(2GM/r)
Where:
- v is the escape velocity
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the celestial body (in this case, the mass of the Sun)
- r is the distance between the object and the center of the celestial body (in this case, the distance between the rocket and the Sun's center, which can vary depending on Earth's orbit)
The mass of the Sun is about 1.989 × 10^30 kilograms. Since we're assuming the rocket is starting from Earth's orbit, the distance between the rocket and the Sun's center would be roughly 149.6 million kilometers (or 93 million miles). However, to perform the calculations, we need to convert the distance to meters.
So, the first step is to convert the distance to meters:
149.6 million kilometers = 149.6 × 10^9 meters
Now we can calculate the escape velocity using the formula mentioned earlier.
v = √(2 × 6.67430 × 10^-11 × 1.989 × 10^30 / 149.6 × 10^9)
After evaluating this equation, we find that the escape velocity from the Sun's gravitational pull at Earth's orbit is approximately 42.1 kilometers per second (about 26.2 miles per second).
Therefore, a rocket would have to be going at least 42.1 kilometers per second (or 26.2 miles per second) to completely leave the solar system starting from Earth's orbit, assuming the Sun is the only mass in the solar system.