I gave this idea some thought some years ago -= catalysed by the events depicted in the video linked at the end.
What you are describing is essentially a rocket.
The concept is obviously possible as it's been done long ago.
Whether it is practical in any real world situations is the main point to be considered.
It is hereby proclaimed that for the balance of this answer.
g = 10 m/s/s
(Multiply this by 0.98 if you wish)
1 kgf is the "weight" of 1 kgm of material in a 1g gravitational field.
So 1 kgm "weighs" 1 kgf = 10 Newton
Units of kgf.s/kgm = s (seconds) !!! :-)
(Only valid for "rocket scientists)
For a rocket to be useful and practical in your scenarios requires it to be "safe", reliably able to be operated instantaneously at any time, suitably powerful and suitably compact and low mass.
There are many potential solutions but one which seems most likely to be a practical contender is a HP (Hydrogen Peroxide) catalytic decomposition rocket. In such a system pressurised HP is expelled via a catalytic decomposition promoter (often a silver wire screen) and hence into a classic convergent divergent rocket motor.
A catalytic HP system has the advantages of being RELATIVELY low temperature, nothing but water and Oyxygen are expelled, it can be stored ready for use without need for cyrogenic cooling and is relatively non corrosive =(although cleanliness is utterly essential). The rocket is able to be "throttled" over most of its range and the thrust can be easily controlled in level and started and stopped at will.
Monopropellant Hydrogen Peroxide is not an especially high energy per volume or mass propellant. The standard "figure of merit" is Isp = specific impile which can be thought of as the amount of time in seconds that a unit mass of propellant can exert a force equal to its own weight in seconds. This can be expressed as eg lbf.s/lbm (pounds force x seconds per pound mass) and a bit of sloppy laziness allows people to cancel the lbf/lbm and call the resulting units "seconds". In fact it's really (lbm. x m/s/s x s / (lbm) = m/s = velocity!
Anon - an Isp of 100 is not too hard and 200 is getting rather hard. Lets goo with 120 to start. 1 kg of HP will provide 120 kg force = say 1200N for 1 second. A little plugging in to formulae shows that you are going to need a significant mass of HP compared with vehicle mass to get very large accelerations. eg your suggested 30 m/s in 3m would (as Olin says) be bone crushing - but in fact is entirely achievable using ABOUT the same amount of propellant as doing it at a more sensible rate - see below. BUT lets look at what it takes to apply 1g of acceleration (10 m/s/s) for 1s to a say 1000 kg car.
1g acceleration to a 1 tone car takes 1000 kg x g = 10,000 N.
Working in kgf, with an Isp of 120 to get 1000 kgf thrust for one second requires 1000/120 = 8.5 kg of HP.
HP density is usefully greater than 1 but call this 8 litres.
Taking the original 30 m/s vehicle - to stop it at 1g requires 30 m/s / 10 m/s/s = 3 seconds of thrust so about 25 kg and 25 litres of HP.
That's about 25/1000 = 2.5% of the vehicle mass - which is not insignificant BUT also 'useful' for something which can either accelerate the vehicle over about 0 - 100 kph in about 3 seconds OR bring it to a standstill in the same distance - even on mud, snow, ice or while in the air.
Above I said that stopping the vehicle in 3 metres was impractical. That's essentially true, but it's not impossible. And a little consideration shows that you notionally need no more propellant to do do As long as you can get the same Isp you just need to consume the same amount of propellant at a higher rate. That requires a higher maximum capacity rocket motor, greater flow rates so bigger "pluming" bigger forces etc- but it's doable.
To stop from 30 m/s (= 108 kph) in 3 metres requires
a = V^2/(2 d) = 900/6 = 150 m/s/s = 15g (as Olin said)
Time to stop is t = V/A = 30/150 = 0.2 seconds.
Doing a 0-100kph ~= 0.60 mph in 0.2 s makes by long long log ago neighbours best toys look like toys indeed - and would make any occupant very sore indeed. BUT not necessarily dead - accelerations of that order over such short periods are survivable given proper containment - and extending it out to say 7.5 metres and 0.5 seconds and 6g has you well inside fighter aircraft and below peak moon-rocket levels.
More useful than applying gross front/back velocity changes might be applying forces to allow steering authority in skids - or for a race car which is sliding across a ploughed surface towards a tyre wall - or a concrete wall. Quite modest amounts of propellant could achieve quite useful results. Under computer control such a system has the ability to extricate a vehicle for otherwise irrecoverable situations. Also, without due care, to insert them into them.
Properly done such a system could even allow a vehicle to "fly" for short periods - like this
(Armadillo Aerospace hydrogen peroxide mono-prop 1st manned flight)