Can we make a rocket with acceleration 17-40 $g$? I have read that humans can survive 17g for several minutes, and 40g for a few fractions of a second. There has been much talk about a 1g rocket and its relativistic capabilities, but even with that a trip to Alpha Centauri would still take more than a year.
What if the rocket slowly accelerates up to 17g for a few minutes, and then 40g for only a few nanoseconds, and then the rate of acceleration stalls to more bearable levels, before ramping back up, over and over again. The human crew would never stay at higher g-forces long enough to kill them, but the ship would gain speed much faster, and more relativist effects would be experienced, making the trip shorter for the crew.
 A: A rocket ride from hell.

What if the rocket slowly accelerates up to 17g for a few minutes

Ouch.  I mean that would be very high for a few minutes.  I suspect it would cause blackouts at best and death or serious injury at worst.  Fighter piolts sustain much lower gees for only very short periods at a time.  And they are exhausting.
To illustrate this try to think about the differece between carrying your normal personal weight and then what would happen your body if I dump 16 more people on top of you as well.  How long would you survive that ?

and then 40g for only a few nanoseconds

In a nanosecond your speed will barely change at all.  It's utterly pointless.  Any period of time for which this actually achieves anything would be death for the crew.

and then the rate of acceleration stalls to more bearable levels, before ramping back up, over and over again.

So essentially you want an extremely violent roller-coaster that can kill for the entire trip.
Not for me thanks.  I suspect volunteers will be in very short supply.
A: It is in principle doable if you are not carrying along the fuel. The only viable option I am aware that would be consistent with known laws of physics are huge laser sails pushed from even bigger orbital laser arrays, capturing and converting TeraWatts of solar energy into coherent Gaussian beams (which have the smallest beam divergence of all coherent wavefronts)
