Ok now I’m confused about something. How can it be the case that an instantaneous perpendicular burn adds to the craft’s speed, but a constant burn just makes it go in a circle with no change in speed?
The trajectory is changing during the continuous burn, so the average direction of the continuous burn is between perpendicular to where the trajectory was at the start of the burn and where it was at the end. The instantaneous burn, by contrast, is assumed to be perpendicular to where the trajectory was at the start only. If you instead made it in between perpendicular to where it was at the start and where it was at the end, as in the continuous burn, you could make it also not add to the craft’s speed.
Going back to the original discussion, yes this means that an instantaneous burn that doesn’t change the speed is pointing slightly forward relative to where the rocket was going at the start of the burn, pushing the rocket slightly backward. But, this holds true even if you have a very tiny exhaust mass sent out at a very high velocity, where it obviously isn’t going at the same speed as the rocket in the planet’s reference frame.
Ok now I’m confused about something. How can it be the case that an instantaneous perpendicular burn adds to the craft’s speed, but a constant burn just makes it go in a circle with no change in speed?
The trajectory is changing during the continuous burn, so the average direction of the continuous burn is between perpendicular to where the trajectory was at the start of the burn and where it was at the end. The instantaneous burn, by contrast, is assumed to be perpendicular to where the trajectory was at the start only. If you instead made it in between perpendicular to where it was at the start and where it was at the end, as in the continuous burn, you could make it also not add to the craft’s speed.
Going back to the original discussion, yes this means that an instantaneous burn that doesn’t change the speed is pointing slightly forward relative to where the rocket was going at the start of the burn, pushing the rocket slightly backward. But, this holds true even if you have a very tiny exhaust mass sent out at a very high velocity, where it obviously isn’t going at the same speed as the rocket in the planet’s reference frame.
I don’t understand what “at the start” is supposed to mean for an event that lasts zero time.
In the case where it’s instantaneous, “at the start” would effectively mean right before (e.g. a one-sided limit).