(a) What is the escape speed on a spherical asteroid whose radius is 435 km and whose gravitational acceleration at the surface is 0.632 m/s2? (b) How far (in km) from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 535 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 2160 km above the surface?
(a) To calculate the escape speed on a spherical asteroid, we can use the formula:
escape speed = √(2 * gravitational acceleration * radius)
Given that the radius of the asteroid is 435 km and the gravitational acceleration at the surface is 0.632 m/s^2, we need to convert the unit of the radius to meters:
radius = 435 km = 435,000 m
Now, we can substitute the values into the formula:
escape speed = √(2 * 0.632 m/s^2 * 435,000 m)
Using a calculator, we can find the escape speed of the asteroid.
(b) To determine how far a particle will go from the surface of the asteroid if it leaves with a radial speed of 535 m/s, we need to consider the gravitational force acting on the particle.
The radial speed is the speed at which the particle is moving away from or towards the center of the asteroid. In this case, we assume the particle is moving away from the surface.
Using the formula of conservation of mechanical energy, we can find the distance traveled by the particle:
Initial mechanical energy = Final mechanical energy
The initial mechanical energy is given by:
Initial mechanical energy = Kinetic energy + Potential energy at the surface
The final mechanical energy is given by:
Final mechanical energy = Kinetic energy + Potential energy at a distance r
Since the particle is leaving the surface of the asteroid, its kinetic energy at the surface is 0, and the potential energy at a distance r is given by:
Potential energy at a distance r = -G * mass of the particle * mass of the asteroid / r
Simplifying these equations and solving for r, we can find the distance traveled by the particle.
(c) To determine the speed at which an object will hit the asteroid when it is dropped from 2160 km above the surface, we need to consider the conservation of mechanical energy.
The initial mechanical energy of the object is given by:
Initial mechanical energy = Kinetic energy + Potential energy at 2160 km above the surface
Since the object was initially at rest, its initial kinetic energy is 0. The potential energy at 2160 km above the surface is given by:
Potential energy at 2160 km above the surface = -G * mass of the object * mass of the asteroid / (radius of the asteroid + 2160 km)
By solving for the final kinetic energy using the conservation of mechanical energy equation and taking the square root, we can find the speed at which the object will hit the asteroid.