(a) What is the escape speed on a spherical asteroid whose radius is 535 km and whose gravitational acceleration at the surface is 3.0 m/s2?
1 ? m/s
(b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 1000 m/s?
2 ? m
(c) With what speed will an object hit the asteroid if it is dropped from 1000 km above the surface?
3 ? m/s
(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 535 km and the gravitational acceleration at the surface is 3.0 m/s², we need to convert the radius to meters before plugging the values into the formula.
First, let's convert the radius from kilometers to meters:
Radius = 535 km * 1000 m/km
Radius = 535,000 m
Now, we can calculate the escape speed:
Escape speed = √(2 * 3.0 m/s² * 535,000 m)
Escape speed ≈ √(3,210,000 m²/s²)
Escape speed ≈ 1793.3 m/s
So, the escape speed on the spherical asteroid is approximately 1793.3 m/s.
(b) To determine how far a particle will go from the surface if it leaves with a radial speed of 1000 m/s, we need to find the distance from the surface where the particle's kinetic energy becomes zero. This distance is called the "turning point" or "maximum height" on our trajectory.
To calculate the maximum height, we can use the conservation of mechanical energy, which states that the sum of the initial kinetic energy and potential energy of the particle should be equal to the sum of its final kinetic energy and potential energy at the highest point:
Initial kinetic energy + Initial potential energy = Final kinetic energy + Final potential energy
The initial potential energy is zero at the surface of the asteroid, and the final kinetic energy is zero at the maximum height. Therefore, we can write the equation as:
1/2 * m * (initial radial speed)² - G * m * M / (radius + h) = 0
where m is the mass of the particle, G is the gravitational constant, M is the mass of the asteroid, radius is the radius of the asteroid, and h is the maximum height.
Rearranging the equation, we get:
h = (m * (initial radial speed)²) / (2 * G * M / (radius + h))
Given that the initial radial speed is 1000 m/s, the radius is 535 km (convert to meters), the gravitational acceleration at the surface is 3.0 m/s² (giving us the acceleration due to gravity, g = 3.0 m/s²), we can find the maximum height.
First, let's convert the radius from kilometers to meters:
Radius = 535 km * 1000 m/km
Radius = 535,000 m
Now, let's plug the values into the equation and solve for h:
h ≈ (m * (1000 m/s)²) / (2 * 6.6743 * 10^-11 m³/(kg * s²) * M / (535,000 m + h))
Unfortunately, without knowing the mass of the particle (m) and the mass of the asteroid (M), we cannot compute the exact value for h.
(c) To find the speed at which an object will hit the asteroid when dropped from 1000 km above the surface, we can use the law of conservation of energy. When the object is at 1000 km above the surface, it has potential energy, and when it hits the surface, this potential energy is converted into kinetic energy.
The potential energy at 1000 km above the surface is given by:
Potential energy = m * g * h
Where m is the mass of the object, g is the gravitational acceleration, and h is the height (1000 km or converted to meters).
Given that the gravitational acceleration at the surface is 3.0 m/s², the height is 1000 km (convert to meters), we can find the potential energy at that height.
First, let's convert the height from kilometers to meters:
Height = 1000 km * 1000 m/km
Height = 1,000,000 m
Now, let's calculate the potential energy:
Potential energy = m * 3.0 m/s² * 1,000,000 m
Potential energy = 3,000,000 m²/s² * m
Since the gravitational potential energy converts into kinetic energy as the object falls, we can equate the potential energy to the kinetic energy just before impact:
Potential energy = Kinetic energy
So, the speed at which the object will hit the asteroid when dropped from 1000 km above the surface will be the square root of the potential energy, which is:
Speed = √(3,000,000 m²/s² * m)
Unfortunately, without knowing the mass of the object (m), we cannot compute the exact value for the speed.