Zero, a hypothetical planet, has a mass of 4.0 1023 kg, a radius of 3.0 106 m, and no atmosphere. A 14 kg space probe is to be launched vertically from its surface.
(a) If the probe is launched with an initial kinetic energy of 5.0 107 J, what will be its kinetic energy when it is 4.0 106 m from the center of Zero?
J
(b) If the probe is to achieve a maximum distance of 8.0 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
J
To solve these problems, we can use the concepts of gravitational potential energy and kinetic energy.
(a) To find the kinetic energy of the space probe when it is 4.0 * 10^6 m from the center of Zero, we need to use the conservation of energy. The initial kinetic energy equals the sum of the final kinetic energy and the change in gravitational potential energy.
The formula for gravitational potential energy is given by:
PE = -G * (m1 * m2) / r
Where G is the gravitational constant (6.67 * 10^-11 m^3 / (kg * s^2)), m1 and m2 are the masses of the objects (in this case, the probe and Zero), and r is the distance between the two objects.
We can first calculate the initial gravitational potential energy when the probe is on the surface of Zero. Since the probe is on the surface, the distance r is equal to the radius of Zero (3.0 * 10^6 m).
PE_initial = -G * (probe_mass * Zero_mass) / (radius_of_Zero)
Next, we can subtract the initial gravitational potential energy from the given initial kinetic energy to find the final kinetic energy.
KE_final = KE_initial - ΔPE
ΔPE = PE_initial - PE_final
ΔPE = 0 - (-G * (probe_mass * Zero_mass) / (4.0 * 10^6 m))
Finally, we can substitute the values into the equation and solve for the final kinetic energy.
KE_final = 5.0 * 10^7 J - [(-G * (probe_mass * Zero_mass) / (4.0 * 10^6 m))]
(b) To find the initial kinetic energy required for the space probe to achieve a maximum distance of 8.0 * 10^6 m from the center of Zero, we can use the same approach.
Again, we first calculate the initial gravitational potential energy:
PE_initial = -G * (probe_mass * Zero_mass) / (radius_of_Zero)
Then, we subtract the final potential energy from the initial potential energy to find the change in gravitational potential energy:
ΔPE = PE_initial - PE_final
ΔPE = PE_initial - (-G * (probe_mass * Zero_mass) / (8.0 * 10^6 m))
Finally, we can add the change in gravitational potential energy to the desired final kinetic energy to find the required initial kinetic energy:
KE_initial = KE_final + ΔPE
KE_initial = 0 + ΔPE