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At 2:16 in this video, Don Pettit says he used the Rocket Equation to determine the velocity of the rocket needed to travel to Mars, which was 16 km/s. I am not sure as how he did it though, and was unsure if he used the escape velocity of Earth, the Hohmann Transfer, and also if that was only that velocity computed needed to be obtained once or through some timeframe. Also, the velocity of the rocket needed to travel to the Moon was 14 km/s, so I was also wondering how much the gravity of the Sun mattered as well.

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16 km/s is about right for Earth surface to low Mars orbit, summing up a few entries in this table.

I am not sure as how he did it though, and was unsure if he used the escape velocity of Earth, the Hohmann Transfer, and also if that was only that velocity computed needed to be obtained once

At a minimum, this would be split into ascent to LEO (~9.4 km/s, done over 10-15 minutes), trans-Mars injection (~4.3 km/s, most likely some hours after reaching LEO), and capture & circularization at Mars arrival (~2.3 km/s, done months later).

Hohmann isn't quite appropriate for the trans-Mars flight, because it doesn't take into account the gravity of Mars -- it applies to going from one circular orbit to another around a single body.

Also, the velocity of the rocket needed to travel to the Moon was 14 km/s, so I was also wondering how much the gravity of the Sun mattered as well.

It's a relatively short step from a translunar trajectory to one that escapes Earth orbit entirely; from there it's only about another 1km/s to do a Mars flyby.

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    $\begingroup$ Perhaps worth noting that these velocity values are delta-v, which represent the amount by which you change the velocity of the rocket. You need to accelerate to ~13.7 km/s to get to the trans-Mars injection, but then you actually need to slow down by ~2.3 km/s to enter Mars orbit. The rocket needs to change its velocity by ~16 km/s in total, but it never actually reaches that speed, since some is spent accelerating and some is spent decelerating. $\endgroup$ – Nuclear Wang Nov 6 '19 at 20:19
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In addition to Russell Borogove's good answer there is another factor here to keep in mind: When you are dealing with planets you can't just add together separate burns. If you add up the energy needed to escape Earth, the energy needed to go from Earth's orbit to Mars' orbit and the energy needed to enter Mars' orbit you will get more than 16 km/sec.

In practice kinetic energy goes at the square of velocity but delta-v doesn't care how fast you're going (unless you're relativistic, at least.) When you are near a planet you are borrowing velocity from it. You normally repay that loan as you leave--but if you burn near the planet the loan doesn't simply end up an even repayment. If you do the trans-Martian injection burn in low Earth orbit it doesn't take much more than escaping Earth would. Likewise, at Mars you both match Mars' orbit and do your capture burn together, as close to Mars as you can. It costs substantially less than doing them separately would. It's call the Oberth effect.

If you have a very low thrust system--say ion or solar sail--you can't get your burn over with near a planet, 16 km/sec will not get you to Mars if your transfer engines are something of this nature. IIRC at Earth it's something like 50% more delta-v needed, I have no idea what it is at Mars.

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