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The principle of conservation of energy is fundamental in physics. It states that in an isolated system, energy cannot be created or destroyed; it can only change forms. In the context of a space probe launched from a planet, this concept simplifies calculations. The total mechanical energy of the probe, which is the sum of its kinetic energy and gravitational potential energy, remains constant throughout its journey if we ignore other influences.
The equation expressing this conservation for the probe is:

• Initial Energy = Final Energy
• \[ K_i + U_i = K_f + U_f \]

Here, \( K_i \) and \( K_f \) are the initial and final kinetic energies, while \( U_i \) and \( U_f \) are the initial and final gravitational potential energies of the probe.
This equation allows us to solve for unknowns, like the final kinetic energy of the probe, by knowing the others. Thus, we can deduce how energy is conserved as the probe travels further from the planet.

Kinetic energy is the energy that an object possesses due to its motion. It is calculated using the equation:

• \[ K = \frac{1}{2} m v^2 \]

where \( m \) is the mass of the object, and \( v \) is its velocity.
In the case of our space probe, its kinetic energy changes as it moves further from the planet Zero. Initially, it has a kinetic energy \( K_i \) given to it at launch. As it moves upwards, some of this kinetic energy converts into potential energy due to the gravitational pull of Zero, slowing the probe down.
As the probe reaches higher altitudes, its velocity decreases, reducing its kinetic energy. According to the conservation of energy formula, as the kinetic energy decreases, the potential energy increases to keep the total energy constant. Understanding this balance is crucial to predicting the motion and energy of moving objects.

The gravitational constant, denoted as \( G \), is a key value in calculating the gravitational forces between two masses. Its numerical value is approximately \( 6.674 \times 10^{-11} \, \text{m}^3\, \text{kg}^{-1}\, \text{s}^{-2} \).
In the context of our problem, \( G \) is crucial for finding the gravitational potential energy of the probe at different distances from the center of the planet Zero. The potential energy \( U \) is given by:

• \[ U = -\frac{G M m}{r} \]

where \( M \) is the mass of the planet, \( m \) is the mass of the probe, and \( r \) is the distance from the center of the planet.
Without the gravitational constant, we couldn't compute how the gravitational attraction affects the potential energy and motion of a space probe. It gives us insight into the universal law of gravitation and its effects on planetary and object interactions.

Mechanical energy is the sum total of kinetic and potential energy in a system. When dealing with the space probe, mechanical energy helps us understand how energy is redistributed as the probe moves.
Mechanical energy can be expressed as:

• \[ E = K + U \]

where \( E \) is the total mechanical energy, \( K \) is the kinetic energy, and \( U \) is the potential energy.
In the absence of external forces, such as air resistance (since Zero has no atmosphere), the mechanical energy of the probe is preserved. This means, for part (a) of our problem, as the probe moves from the surface to a height of \( 4.0 \times 10^6 \, \text{m} \), its mechanical energy remains unchanged.
By understanding and calculating mechanical energy, we can predict how much kinetic energy remains as the potential energy increases. This insight helps in planning space missions by providing the needed initial energies to achieve desired altitudes or speeds.