To determine the minimum velocity required for an object to leave the solar system from Earth's surface, we need to consider the gravitational potential energy and kinetic energy of the object.
The minimum velocity required to escape the gravitational pull of the Sun can be calculated using the concept of escape velocity. The escape velocity from Earth's surface is given by 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 Sun,
- r is the distance between the object and the center of the Sun.
In this case, we can assume that the object starts at the surface of the Earth, so the distance r will be the sum of the radius of the Earth's orbit and the radius of the Earth itself.
r = 1.5 × 10^11 m + 6.4 × 10^6 m.
Now, we can calculate the escape velocity:
v = √(2 × 6.67430 × 10^(-11) m^3⋅kg^(-1)⋅s^(-2) × 2 × 10^30 kg / (1.5 × 10^11 m + 6.4 × 10^6 m)).
Simplifying this equation will give us the minimum escape velocity from Earth's surface:
v ≈ 42,122 m/s.
Therefore, the minimum velocity relative to the Earth's surface required for an object to escape the solar system is approximately 42,122 m/s.