Final answer:
The escape velocity from the surface of a planet of mass M and radius R can be calculated using the formula v₁ = sqrt(2GM/R), where G is the gravitational constant. The escape velocity is the minimum velocity needed for an object to escape the gravitational pull of the planet. It does not depend on the object's mass and remains the same beyond the surface of the planet.
Step-by-step explanation:
To find the escape velocity from the surface of a planet of mass M and radius R, we set the total energy equal to zero. At the surface of the planet, the object has escape velocity v₁ and reaches infinity with velocity v₂ = 0. Substituting into the equation, we have GMm/R = 1/2mv₁², which can be simplified to v₁ = sqrt(2GM/R).
The escape velocity does not depend on the object's mass, meaning it is the same for all objects regardless of their mass. The escape velocity is the minimum velocity needed for an object to escape the gravitational pull of the planet. Beyond the surface of the planet, the escape velocity remains the same.
For example, if we consider the mass of Earth (M = 5.97 × 10^24 kg) and its radius (R = 6,371 km), we can calculate the escape velocity using the formula v₁ = sqrt(2GM/R) = sqrt(2 * (6.67 × 10^(-11) Nm²/kg²) * (5.97 × 10^24 kg) / (6,371,000 m)) ≈ 11.2 km/s.