Zero, a hypothetical planet, has a mass of 5.7 x 1023 kg, a radius of 2.8 x 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 x 107 J, what will be its kinetic energy when it is 4.0 x 106 m from the center of Zero? (b) If the probe is to achieve a maximum distance of 8.0 x 106 m from the center of Zero, with what initial kinetic energy must it be launched from the surface of Zero?
To solve this problem, we can use the conservation of energy principle. The total mechanical energy of the probe is conserved throughout its motion.
(a) To find the kinetic energy when the probe is at a distance of 4.0 x 10^6 m from the center of Zero, we can use the conservation of energy principle:
The initial kinetic energy of the probe is given as 5.0 x 10^7 J. At this distance, we can assume the gravitational potential energy is negligible compared to the kinetic energy. Therefore, the total mechanical energy is conserved:
Initial kinetic energy = Final kinetic energy
Using this equation and the values given in the problem:
5.0 x 10^7 J = Final kinetic energy
(b) To 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 again use the conservation of energy principle:
At the maximum distance, the gravitational potential energy is maximum, and the kinetic energy is minimum. Therefore, the total mechanical energy remains constant:
Initial kinetic energy + Initial gravitational potential energy = Final kinetic energy + Final gravitational potential energy
The initial gravitational potential energy is equal to the product of the gravitational constant, the probe's mass, and the mass of Zero divided by the initial distance from the center of Zero:
Initial gravitational potential energy = (G * mass of probe * mass of Zero) / initial distance from Zero's center
The final gravitational potential energy is equal to the product of the gravitational constant, the probe's mass, and the mass of Zero divided by the final distance from the center of Zero:
Final gravitational potential energy = (G * mass of probe * mass of Zero) / final distance from Zero's center
Using the values given in the problem, and rearranging the equation:
Initial kinetic energy = Final kinetic energy + Initial gravitational potential energy - Final gravitational potential energy
and
Initial gravitational potential energy = (G * mass of probe * mass of Zero) / initial distance from Zero's center
Final gravitational potential energy = (G * mass of probe * mass of Zero) / final distance from Zero's center
We can substitute the values into the equations, and calculate the final result.
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the sum of the kinetic energy and potential energy of an object remains constant if the only external forces acting on it are conservative.
(a) To find the kinetic energy of the space probe when it is 4.0 x 10^6 m from the center of Zero, we need to determine its initial potential energy. We can calculate the initial potential energy as:
Potential energy = mass x gravitational field strength x height
Given:
Mass of Zero (M) = 5.7 x 10^23 kg
Radius of Zero (R) = 2.8 x 10^6 m
Mass of the probe (m) = 10 kg
Initial kinetic energy (Ki) = 5.0 x 10^7 J
Final distance from the center of Zero (r) = 4.0 x 10^6 m
The gravitational field strength (g) at the surface of Zero can be calculated using the formula:
g = GM/R^2
where G is the gravitational constant (6.67430 x 10^-11 N m^2/kg^2). Thus:
g = (6.67430 x 10^-11 N m^2/kg^2)(5.7 x 10^23 kg) / (2.8 x 10^6 m)^2
Now, we can calculate the initial potential energy (Pi) of the probe using the formula:
Potential energy = mass x gravitational field strength x height
Pi = mgh
= (10 kg)(g)(R)
Note that the potential energy is negative because we are considering it to be zero at an infinite distance away from Zero. Therefore, Pi = -10 g R.
The initial total mechanical energy (Ei) can be calculated as the sum of the initial kinetic energy and the initial potential energy:
Ei = Ki + Pi
= 5.0 x 10^7 J + (-10 g R) J
To find the final kinetic energy (Kf) when the probe is at a distance of 4.0 x 10^6 m from the center of Zero, we can use the conservation of mechanical energy:
Ei = Ef
Since the only forces acting on the probe are conservative (gravity), the total mechanical energy remains constant.
Now, the final potential energy (Pf) can be calculated as the product of the mass, the gravitational field strength, and the final distance:
Pf = mgh
= (10 kg)(g)(r)
where r is the final distance from the center of Zero.
Therefore, the final kinetic energy (Kf) is given by:
Kf = Ef - Pf
= Ei - Pf
Substituting the values:
Kf = (5.0 x 10^7 J + (-10 g R) J) - [(10 kg)(g)(4.0 x 10^6 m)]
(b) To 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 can use the same approach as in part (a).
We need to find the potential energy when the probe is at a distance of 8.0 x 10^6 m from the center of Zero. Similarly, the final kinetic energy can be calculated using the conservation of mechanical energy.
Pf = mgh
= (10 kg)(g)(8.0 x 10^6 m)
Kf = Ei - Pf
Once you substitute the values, you will get the final result.
Remember to use the correct units and value for the gravitational constant (G) in your calculations.