Applicability
The rocket equation captures the essentials of rocket flight physics in a single short equation. It also holds true for rocket-like reaction vehicles whenever the effective exhaust velocity is constant; and can be summed or integrated when the effective exhaust velocity varies. It does not apply to non-rocket systems, such as aerobraking, gun launches, space elevators, launch loops, tether propulsion.
Delta-v is of fundamental importance in orbital mechanics. It quantifies how difficult it is to perform a given orbital maneuver. To achieve a large delta-v, either must be huge (growing exponentially as delta-v rises), or must be tiny, or must be very high, or some combination of all of these.
In practice, very-high delta-v has been achieved by a combination of 1) very large rockets (increasing ), 2) staging (decreasing ), and 3) very high exhaust velocities.
The Saturn V rocket used in the Apollo space program is an example of a large, serially staged rocket. The Space Shuttle is an example of parallel staging where all of its engines are ignited on the ground and some (the solid rocket boosters) are jettisoned to lose weight before reaching orbit.
The ion thruster is an example of a high exhaust velocity rocket. Instead of storing energy in the propellant itself as in a chemical rocket, ion and other electric rockets separate energy storage from the reaction (propellant) mass storage. Not only does this allow very large (and in principle unlimited) amounts of energy to be applied to small amounts of ejected mass to achieve very high exhaust velocities, but energy sources far more compact than chemical fuels can be used, such as nuclear reactors. In the inner solar system solar power can be used, entirely eliminating the need for a large internal primary energy storage system.
Read more about this topic: Tsiolkovsky Rocket Equation