Apply the Energy Principle. Define the system to be the space probe and Sun.
Neglect interactions of the system with its surroundings (especially Earth and Jupiter). Thus, there is no work done on the system by the surroundings, and the energy of the system is constant.
The energy of the system includes the particle energies of the space probe and Sun and their interaction energy (gravitational potential energy). Define the initial state of the system to be when the space probe is at perihelion (near Earth) and the final state of the system to be when the space probe is at aphelion (near Jupiter). Thus, applying the energy principle
Substitute expressions for the particle energies in terms of rest energy and kinetic energy.
The particle energy includes both rest energy and kinetic energy. But the final rest energy of the system is the same as the initial rest energy of the system (assuming that neither the probe nor Sun loses or gains significant mass during this process). Therefore, the rest energy cancels. Also, the kinetic energy of Sun is negligible. It's so massive compared to the probe (and the planets) for that matter that is moves very slightly in its own orbit. This is called a wobble, but is negligible.
Solve for the initial kinetic energy. Substitute the known values. Note that the final distance of the probe from Sun is at the orbital radius of Jupiter and the initial position of the probe from Sun is at the orbital radius of Earth.
The mass of the space probe m is not known; therefore, we cannot get a numerical answer for the initial kinetic energy. But we can still solve for the initial speed by substituting an expression for the initial kinetic energy.
Divide both sides by m and solve for the final speed of the probe.
Substitute , , , , and . Then,
The initial speed of the probe when leaving Earth ( ) is greater than its speed when it arrives at Jupiter ( ). This makes sense because as it travels to Jupiter, the system gains potential energy and therefore loses kinetic energy.
