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It's possible to do a ME without a BE (or BS-Eng) first. For example Boston University offers such a program, the "Late Entry Accelerated Program" (LEAP) Masters in Engineering, with 7 possible flavors of engineering. I know there are others, just not sure who has MechE. In any case, I would suggest a heavy focus on math/sci in your undergrad - as much calculus as you can stand, all the way through multivar and differential equations, statistics (as opposed to statics, but that would be good to have, too) plus physics 1 & 2. If you have a community college nearby, you may be able to get that statics class, plus maybe dynamics, fluid mechanics, heat transfer...some of the other undergrad stuff that a MechE would study...even if your actual bachelors is not engineering. That will demonstrate that you're a) interested, and b) capable. Use your electives (both gen eds and free electives) to further your goal...it'll make the transition to ME that much easier. (A friend of mine did an undergrad in Industrial Hygiene, then a law degree, and *then* a MS in Computer Engineering. So, it *can* be done.)
Start working on your gen-eds as you research the ME options that interest you. Contact program directors and/or admissions counselors to discuss your specific goals and interests. You may find that you want a hands-on mechanical engineering program, or that you enjoy engineering management, or any other options. Get specifics - "If I have degree A, what is my path to a spot in your ME?"
The TESU Nuke program is an interesting option, but you'll spend a lot of time on things not relevant to general MechE. Also, it's too light on math, and doesn't get into design at all. The BSAST in Technical Studies will give you room to take the heat transfer, etc, without wasting so much time on the reactor-heavy side-track. Plus, you can get plenty more math, and even squeeze in a design course at a local college, within the Technical Studies framework. I think that the more of a foundation you can build in your undergrad, the better your chance of success in the ME. You're going to have to do the math, etc, anyway, in order to understand the masters level MechE work, so might as well get it out of the way as part of a degree than as extra stuff *after* your bachelors, delaying your start in "the good stuff."
ETA - I notice that you mentioned the BSBA. In your specific case, I think that degree might also side-track you a little. Yes, it's super fast and super cheap, but I don't know if it allows you enough wiggle room to get all of the math/phys you should have, plus you'll be spending precious time on business studies and not engineering/math/sci. If you can slam through it, then the BSBA may work. But I would hesitate to spend so much time/energy on so many credits worth of stuff that kinda won't matter in pure engineering. (Now, engineering management, that's another story. The BSBA could help there.) I do think the BSAST shows more deliberate preparation for the ME, vs the BSBA, at first glance. Those ME admissions counselors can offer insight there.
It's possible to do a ME without a BE (or BS-Eng) first. For example Boston University offers such a program, the "Late Entry Accelerated Program" (LEAP) Masters in Engineering, with 7 possible flavors of engineering. I know there are others, just not sure who has MechE. In any case, I would suggest a heavy focus on math/sci in your undergrad - as much calculus as you can stand, all the way through multivar and differential equations, statistics (as opposed to statics, but that would be good to have, too) plus physics 1 & 2. If you have a community college nearby, you may be able to get that statics class, plus maybe dynamics, fluid mechanics, heat transfer...some of the other undergrad stuff that a MechE would study...even if your actual bachelors is not engineering. That will demonstrate that you're a) interested, and b) capable. Use your electives (both gen eds and free electives) to further your goal...it'll make the transition to ME that much easier. (A friend of mine did an undergrad in Industrial Hygiene, then a law degree, and *then* a MS in Computer Engineering. So, it *can* be done.)
Start working on your gen-eds as you research the ME options that interest you. Contact program directors and/or admissions counselors to discuss your specific goals and interests. You may find that you want a hands-on mechanical engineering program, or that you enjoy engineering management, or any other options. Get specifics - "If I have degree A, what is my path to a spot in your ME?"
The TESU Nuke program is an interesting option, but you'll spend a lot of time on things not relevant to general MechE. Also, it's too light on math, and doesn't get into design at all. The BSAST in Technical Studies will give you room to take the heat transfer, etc, without wasting so much time on the reactor-heavy side-track. Plus, you can get plenty more math, and even squeeze in a design course at a local college, within the Technical Studies framework. I think that the more of a foundation you can build in your undergrad, the better your chance of success in the ME. You're going to have to do the math, etc, anyway, in order to understand the masters level MechE work, so might as well get it out of the way as part of a degree than as extra stuff *after* your bachelors, delaying your start in "the good stuff."
ETA - I notice that you mentioned the BSBA. In your specific case, I think that degree might also side-track you a little. Yes, it's super fast and super cheap, but I don't know if it allows you enough wiggle room to get all of the math/phys you should have, plus you'll be spending precious time on business studies and not engineering/math/sci. If you can slam through it, then the BSBA may work. But I would hesitate to spend so much time/energy on so many credits worth of stuff that kinda won't matter in pure engineering. (Now, engineering management, that's another story. The BSBA could help there.) I do think the BSAST shows more deliberate preparation for the ME, vs the BSBA, at first glance. Those ME admissions counselors can offer insight there.