Terminal velocity isn't the fastest that an object can travel. It's just the speed at which air resistance and gravity reach an equilibrium. So it's 100% possible to accelerate something past its terminal velocity. Whether you're capable of hurtling a pissed off ball of teeth and claws past that threshold is more a question of your abilities than it is of physics.
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IDK; what do you look like and how active are you? Terminal velocity of a squirrel is about 20mph. If you can throw a squirrel at or above that, you succeed. The squirrel being alive or dead may also have an appreciable effect on the outcome. Example, if alive it may bite or scratch you while attempting to throw it and make you fuck up.
Animal cruelty apart, you could probably increase its terminal velocity by dipping it in water first. This would increase its mass and reduce air resistance.
Some dude just dunking a pissed off squirrel in a five gallon bucket and hucking it as hard as he can.
0_o
Please don't, but as a follow-up question: If you throw something downward, would it usually exceed terminal velocity?
Depends. If you can throw something faster than it's terminal velocity then it will exceed it until the air resistance will brake it.
Imagine throwing a balloon downwards. You'll have to exceed it's terminal velocity just to move it, and it also quickly stops going faster.
Same thing happens to other stuff. It can move faster than terminal velocity, but the force you apply will not be permanently added to its speed.
Imagine throwing a bowling ball downwards. It has a high terminal velocity. You can't throw it that fast. By throwing it downwards, you are helping it achieve terminal velocity, but it doesn't exceed it when it reaches it.