Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.

How can the energy saved from the Oberth effect exceed the chemical energy stored in the rocket?

Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.

Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding 100 m/s while initially travelling at 1000 m/s will add a great deal more energy to the ship than adding 100 m/s while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to Kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.

How can the energy saved from the Oberth effect exceed the chemical energy stored in the rocket?

Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding $100\ \text{m/s}$ while initially travelling at $1000\ \text{m/s}$ will add a great deal more energy to the ship than adding $100 \ \text{m/s}$ while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.

How can the energy saved from the Oberth effect exceed the chemical energy stored in the rocket?

I'm asking here to clear up a potential misconception or have an explanation on how exactly the oberth effect works. context: rocket in deep space, no external forces on the rocket. From what I understand, using the formula for linear kinetic energy KE = m(v^2)/2 you can conclude that adding 100m/s while initially travelling at 1000m/s will add a great deal more energy to the ship than adding 100m/s while initially at rest after the initial energy is accounted for: (((1000+100)^2) - (1000^2))*M/2 >> (100^2)*M/2

And, from what I understand of Tsiolkovsky's rocket equation, this velocity I am supposedly adding is proportional to the exhaust velocity of the propellant relative to the craft, and the natural logarithm of the ratio of mass of propellant used.

In both cases where the craft is at rest or at an initial velocity, the ∆v is the same, and is dependant on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this ∆v. So I'd wager the same quantity of chemical energy is converted to Kinetic energy, yet I'm left with this seemingly unexplained (in the example) 200,000 (21001000) J/Kg more energy, and I'm clueless as to where it could have come from.

I've searched around the internet, resulting in explanations involving efficiency, but what about when the energy supposedly saved from the oberth effect exceeds the chemical energy stored in the rocket?

Consider a rocket in deep space with no external forces. Using the formula for linear kinetic energy $$\text{KE} = mv^2/2$$ we find that adding 100 m/s while initially travelling at 1000 m/s will add a great deal more energy to the ship than adding 100 m/s while initially at rest: $$(1100^2 - 1000^2) \frac{m}{2} \gg (100^2) \frac{m}{2}.$$ In both cases, the $\Delta v$ is the same, and is dependent on the mass of fuel used, hence the same mass and number of molecules is used in the combustion process to obtain this $\Delta v$. So I'd wager the same quantity of chemical energy is converted to Kinetic energy, yet I'm left with this seemingly unexplained $200,000\ \text{J/kg}$ more energy, and I'm clueless as to where it could have come from.

How can the energy saved from the Oberth effect exceed the chemical energy stored in the rocket?