Zero, a hypothetical planet, has a mass of 2.7 x 10^23 kg, a radius of 2.8 x 10^6 m, and no atmosphere. A 3.6 kg space probe is to be launched vertically from its surface. (a) If the probe is launched with an initial kinetic energy of 8.0 × 10^7 J, what will be its kinetic energy when it is 3.7 × 10^6 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 4.6 × 10^6 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
Intial total energy= final total energy
Initial KE+INitial PE=KE2+PE2
where ke2= 1/2 mv^r at some r, and pe2=GMm/r
To solve this problem, we can use the principle of conservation of energy. The total mechanical energy (potential energy + kinetic energy) of the space probe will remain constant throughout its motion.
(a) To find the kinetic energy of the probe when it is 3.7 × 10^6 m from the center of Zero, we can use the equation for mechanical energy:
Total mechanical energy = Potential energy + Kinetic energy
At the surface of Zero, the potential energy is zero because it is defined as the reference point. Therefore, the total mechanical energy is equal to the initial kinetic energy:
Total mechanical energy = 8.0 × 10^7 J
When the space probe is at a distance of 3.7 × 10^6 m from the center of Zero, we can calculate the potential energy using the equation:
Potential energy = GMm/r
where G is the gravitational constant, M is the mass of Zero, m is the mass of the probe, and r is the distance between the center of Zero and the probe.
Plugging in the values:
Potential energy = (6.67 × 10^-11 N m^2/kg^2) × (2.7 × 10^23 kg) × (3.6 kg) / (3.7 × 10^6 m)
Now we can subtract the potential energy from the total mechanical energy to find the kinetic energy:
Kinetic energy = Total mechanical energy - Potential energy
(b) To find the initial kinetic energy required for the probe to achieve a maximum distance of 4.6 × 10^6 m from the center of Zero, we can use the same approach:
Total mechanical energy = Potential energy + Kinetic energy
At a maximum distance from Zero, the potential energy is zero because it is defined as the reference point. Therefore, the total mechanical energy is equal to the maximum kinetic energy we want to find.
When the space probe is at a distance of 4.6 × 10^6 m from the center of Zero, we can calculate the potential energy in the same way as before:
Potential energy = GMm/r
Plugging in the values:
Potential energy = (6.67 × 10^-11 N m^2/kg^2) × (2.7 × 10^23 kg) × (3.6 kg) / (4.6 × 10^6 m)
Again, we can subtract the potential energy from the total mechanical energy to find the initial kinetic energy.