(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?
To answer these questions, we need to use the concept of escape speed, which is the minimum speed an object needs to escape the gravitational pull of a celestial body.
(a) To find 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:
1. Convert the radius to meters: 435 km * 1000 m/km = 435,000 m
2. Substitute the values into the formula:
escape speed = √(2 * 0.632 m/s^2 * 435,000 m)
= √(2 * 0.632 * 435,000) m/s
Now you can calculate the escape speed on the spherical asteroid.
(b) To find how far a particle will go from the surface if it leaves the asteroid's surface with a radial speed of 535 m/s, we need to consider the effects of both the gravitational pull and the initial velocity.
1. Start by finding the escape speed using the formula from part (a).
2. Compare the radial speed of the particle (535 m/s) with the escape speed. If the radial speed is greater than the escape speed, the particle will escape from the asteroid.
To calculate the distance from the surface the particle will go, we can use the concept of energy conservation:
Total mechanical energy = (1/2) * (radial speed)^2 - (gravitational potential energy)
The gravitational potential energy is given by:
gravitational potential energy = - (gravitational constant * mass of the particle * mass of the asteroid) / (radius of the asteroid)
Since we don't have information about the mass of the particle or the asteroid, we can't calculate the precise distance. However, we can still calculate it relative to the radius of the asteroid:
distance from the surface = (total mechanical energy) / (gravitational potential energy per unit distance)
(c) To find the speed at which an object will hit the asteroid when it is dropped from a certain height above the surface (2160 km in this case), we can use the principle of energy conservation.
The total mechanical energy of the object can be given by:
Total mechanical energy = (1/2) * (falling speed)^2 + (gravitational potential energy)
We know that the object is dropped, so the initial speed is 0 m/s. Therefore:
Total mechanical energy = (1/2) * (0)^2 + (gravitational potential energy)
The gravitational potential energy can be given by:
gravitational potential energy = - (gravitational constant * mass of the object * mass of the asteroid) / (distance from the center of the asteroid)
The distance from the center of the asteroid can be calculated as the sum of the radius of the asteroid and the height from the surface.