(a) What is the escape speed on a spherical asteroid whose radius is 503 km and whose gravitational acceleration at the surface is 0.464 m/s2? (b) How far from the surface will a particle go if it leaves the asteroid's surface with a radial speed of 523 m/s? (c) With what speed will an object hit the asteroid if it is dropped from 920.5 km above the surface?
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To find out how far from the surface a particle will go if it leaves the asteroid's surface with a radial speed of 523 m/s, we can use the conservation of energy principle.
The conservation of energy principle states that the sum of kinetic energy and potential energy remains constant in a closed system. In this case, as the particle moves away from the asteroid's surface, it will lose kinetic energy and gain potential energy.
Let's go step by step to find the answer:
Step 1: Calculate the gravitational potential energy at the surface of the asteroid.
The potential energy at the surface of the asteroid can be calculated using the formula:
PE = mgh
Where:
PE = potential energy
m = mass (we'll assume the value to be 1 kg for simplicity)
g = acceleration due to gravity at the surface of the asteroid (0.464 m/s^2)
h = height or distance above the surface of the asteroid (0, as we are calculating at the surface)
PE = 1 kg * 0.464 m/s^2 * 0 m
PE = 0 J (Joules)
Step 2: Calculate the kinetic energy at the surface of the asteroid.
The kinetic energy at the surface of the asteroid can be calculated using the formula:
KE = (1/2)mv^2
Where:
KE = kinetic energy
m = mass (again, we'll assume the value to be 1 kg for simplicity)
v = radial speed of the particle (523 m/s)
KE = (1/2) * 1 kg * (523 m/s)^2
KE = 136,409.5 J (Joules)
Step 3: Calculate the total mechanical energy at the surface of the asteroid.
The total mechanical energy at the surface of the asteroid is the sum of the potential energy and kinetic energy.
Total mechanical energy = PE + KE
Total mechanical energy = 0 J + 136,409.5 J
Total mechanical energy = 136,409.5 J (Joules)
Step 4: Calculate the potential energy at the new distance.
Since the total mechanical energy remains constant, we can equate the initial total mechanical energy to the potential energy at the new distance to find it.
Total mechanical energy = potential energy at new distance
Potential energy at new distance = Total mechanical energy
Potential energy at new distance = 136,409.5 J (Joules)
Step 5: Calculate the new distance.
Now we can rearrange the potential energy formula and solve for the new distance (h).
Potential energy = mgh
h = Potential energy / (mg)
h = 136,409.5 J / (1 kg * 0.464 m/s^2)
h ≈ 294,388.4 m
Therefore, if the particle leaves the asteroid's surface with a radial speed of 523 m/s, it will go approximately 294,388.4 meters away from the surface of the asteroid.