To determine the escape speed of a particle placed at a distance of R/2 from the center of the Earth, we first need to understand the concept of escape velocity. Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without any additional propulsion. In this case, we will derive the escape speed using gravitational principles.
Understanding Gravitational Forces
The gravitational force acting on the particle can be described by Newton's law of gravitation. The force experienced by the particle of mass m at a distance r from the center of the Earth is given by:
F = G * (M * m) / r²
Here, G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth. When the particle is at R/2, we can substitute r with R/2:
F = G * (M * m) / (R/2)² = (4G * M * m) / R²
Potential Energy at R/2
The gravitational potential energy (U) of the particle at a distance r from the center of the Earth is given by:
U = -G * (M * m) / r
Substituting r with R/2, we find:
U = -G * (M * m) / (R/2) = -2G * (M * m) / R
Calculating Escape Velocity
To escape the gravitational pull, the particle must have enough kinetic energy (K.E.) to overcome the gravitational potential energy. The kinetic energy of the particle is given by:
K.E. = (1/2) * m * v²
Setting the kinetic energy equal to the magnitude of the potential energy (since we need to overcome it), we have:
(1/2) * m * v² = 2G * (M * m) / R
We can simplify this equation by canceling out the mass m from both sides:
(1/2) * v² = 2G * M / R
Multiplying both sides by 2 gives:
v² = 4G * M / R
Taking the square root of both sides, we find the escape speed v:
v = √(4G * M / R)
Final Expression for Escape Speed
This expression tells us that the escape speed from a distance of R/2 from the center of the Earth is dependent on the gravitational constant G, the mass of the Earth M, and the radius of the Earth R. To put this into perspective, if we plug in the values for G (approximately 6.674 × 10⁻¹¹ N(m/kg)²), M (approximately 5.972 × 10²⁴ kg), and R (approximately 6.371 × 10⁶ m), we can calculate the numerical value of the escape speed.
In summary, the escape speed of a particle placed at R/2 from the center of the Earth can be derived using gravitational principles, leading us to the formula v = √(4G * M / R). This illustrates the interplay between gravitational forces and kinetic energy in determining the conditions necessary for an object to escape a planet's gravitational influence.
