is lost as waste heat in the exhaust that the nozzle was unable to extract. The required energy is Conclusions: for we have for a given , the minimum energy is needed if , requiring an energy of . In the case of acceleration in a fixed direction, and starting from zero speed, and in the absence of other forces, this is 54.4% more than just the final kinetic energy of the payload. In this optimal case the initial mass is 4.92 times the final mass. These results apply for a fixed exhaust speed. Due to the Oberth effect and starting from a nonzero speed, the required potential energy needed from the propellant may be less than the increase in energy in the vehicle and payload. This can be the case when the reaction mass has a lower speed after being expelled than before – rockets are able to liberate some or all of the initial kinetic energy of the propellant. Also, for a given objective such as moving from one orbit to another, the required may depend greatly on the rate at which the engine can produce and maneuvers may even be impossible if that rate is too low. For example, a launch to Low Earth orbit (LEO) normally requires a of ca. 9.5 km/s (mostly for the speed to be acquired), but if the engine could produce at a rate of only slightly more than g, it would be a slow launch requiring altogether a very large (think of hovering without making any progress in speed or altitude, it would cost a of 9.8 m/s each second). If the possible rate is only or less, the maneuver can not be carried out at all with this engine. The power is given by where is the thrust and the acceleration due to it. Thus the theoretically possible thrust per unit power is 2 divided by the specific impulse in m/s. The thrust efficiency is the actual thrust as percentage of this. If, e.g., solar power is used, this restricts ; in the case of a large the possible acceleration is inversely proportional to it, hence the time to reach a required delta-v is proportional to ; with 100% efficiency: for we have Examples: power, 1000W; mass, 100 kg; = 5 km/s, = 16 km/s, takes 1.5 months. power, 1000W; mass, 100 kg; = 5 km/s, = 50 km/s, takes