Final answer:
To find the height to which a particle rises above the surface of the Earth when projected vertically upwards with a kinetic energy equal to half of the minimum value needed for escape, we can consider the conservation of energy.
Using this principle, we can derive the equation for the height as the radius of the Earth multiplied by an expression involving the gravitational constant, the mass of the Earth, and the total energy required for escape.
Step-by-step explanation:
To find the height to which the particle rises above the surface of the Earth, we can consider the conservation of energy. Given that the kinetic energy of the particle is equal to half of the minimum value needed for it to escape, we can equate the kinetic energy to the gravitational potential energy at the highest vertical position.
Let's assume the minimum escape velocity is ve. The kinetic energy of the particle at the highest point is equal to (1/2)mv^2, where m is the mass of the particle and v is the velocity of the particle at the highest point. The gravitational potential energy at the highest point is equal to GmM/ r, where G is the gravitational constant, M is the mass of the Earth, and r is the radius of the Earth.
Setting the kinetic energy equal to the gravitational potential energy, we have (1/2)mv^2 = GmM/ r. Solving for v, we get v = sqrt(2GM/ r).
From the conservation of energy, we know that the sum of the kinetic energy and the gravitational potential energy remains constant throughout the motion. So, at the highest point, the kinetic energy is zero, and the gravitational potential energy is equal to the initial total energy. So, we have GmM/ r = Einitial.
Since the kinetic energy is half of the minimum value needed for escape, the initial total energy is equal to Eescape/2, where Eescape is the total energy required for escape. Therefore, we have GmM/ r = Eescape/2. Rearranging the equation, we get r = (2GM/ Eescape)^(1/2).
So, the height to which the particle rises above the surface of the Earth is equal to the radius of the Earth multiplied by the expression (2GM/ Eescape)^(1/2).