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?)

To break free of Earth's gravity and enter space, a rocket needs to reach the escape velocity, which is about 11.2 kilometers per second (or 40,270 kilometers per hour). This speed accounts for the gravitational pull of Earth and allows the rocket to overcome it.

To completely leave the solar system, however, we need to consider the gravitational pull of the Sun. The escape velocity from the Sun's gravitational field can be calculated using the same principles as escaping Earth's gravity.

The equation to calculate the escape velocity is:

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 Sun (approximately 1.989 × 10^30 kg)
- r is the distance between the rocket and the center of the Sun

Considering the rocket begins its journey from Earth's orbit, which is approximately 149.6 million kilometers away from the Sun (or 1 AU), we can calculate the escape velocity from the solar system.

First, convert the distance from kilometers to meters:
r = 149.6 million km * 1000 m/km = 1.496 × 10^11 meters

Next, substitute the values into the equation to solve for v:
v = √(2 * (6.67430 × 10^(-11) m^3 kg^(-1) s^(-2)) * (1.989 × 10^30 kg) / (1.496 × 10^11 meters))

By calculating this equation, you can find the speed required for a rocket to completely leave the solar system.