Zero, a hypothetical planet, has a mass of 1.0 1023 kg, a radius of 3.0 106 m, and no atmosphere. A 10 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?
(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?
Use the rel;ationship
KE + Potential energy) = constant.
The potential energy at distance R from the center is
-GMm/R
M is the planet's mass and m is the probes.
That means (1/2) mV^2 - GMm/R = constant
Use that fact to compute the unknown kinetic energy in each problem
To solve both parts of the problem, we need to use the relationship between kinetic energy, potential energy, and the distance from the center of Zero.
The potential energy at a distance R from the center of Zero is given by the formula -GMm/R, where G is the gravitational constant, M is the mass of Zero, and m is the mass of the space probe.
The total mechanical energy, which includes both kinetic and potential energy, is conserved and remains constant throughout the motion of the probe. Therefore, we can write the equation as follows:
(1/2)mv^2 - GMm/R = constant
Now let's solve the two parts of the problem:
(a) If the probe is launched with an initial kinetic energy of 5.0 x 10^7 J and we want to find its kinetic energy when it is 4.0 x 10^6 m from the center of Zero.
Let's denote the initial distance from the center of Zero as R1 and the final distance as R2. Therefore, we have:
(1/2)mv1^2 - GMm/R1 = (1/2)mv2^2 - GMm/R2
Since the total mechanical energy is conserved, we can set the two expressions equal to each other:
(1/2)mv1^2 - GMm/R1 = (1/2)mv2^2 - GMm/R2
Now, substituting the given values:
(1/2)(10 kg)(v1)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (3.0 x 10^6 m) = (1/2)(10 kg)(v2)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (4.0 x 10^6 m)
Simplifying and solving for v2:
(1/2)(10 kg)(v1)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (3.0 x 10^6 m) + (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (4.0 x 10^6 m) = (1/2)(10 kg)(v2)^2
Now you can solve for v2 by rearranging the equation and plugging in the given values.
(b) Now, let's find the initial kinetic energy required for the probe to achieve a maximum distance of 8.0 x 10^6 m from the center of Zero.
We need to find the initial kinetic energy (KE1) that will correspond to a maximum distance of R (8.0 x 10^6 m). Therefore, we have:
(1/2)m(v1)^2 - GMm/R1 = (1/2)m(v2)^2 - GMm/R
Again, since the total mechanical energy is conserved, we can set the two expressions equal to each other:
(1/2)m(v1)^2 - GMm/R1 = (1/2)m(v2)^2 - GMm/R
Now, substituting the given values:
(1/2)(10 kg)(v1)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (3.0 x 10^6 m) = (1/2)(10 kg)(v2)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (8.0 x 10^6 m)
Simplifying and solving for KE1:
(1/2)(10 kg)(v1)^2 - (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (3.0 x 10^6 m) + (6.67 x 10^-11 N*m^2/kg^2)(1.0 x 10^23 kg)(10 kg) / (8.0 x 10^6 m) = (1/2)(10 kg)(v2)^2
Now you can solve for KE1 by rearranging the equation and plugging in the given values.