Could somebody survive a fall by jumping off an object shortly before impacting? It’s hard to explain what I’m asking for, so I made a diagram:



*

*The person is falling.

*The person stands on a large object while falling (let’s say a part of a plane).

*Just before the objects hits the ground, the person jumps with great force.

*The person lands safely on the ground after he jumps.
Would this be possible for a person to do? I know this is a physics related question, so assume that the person’s mass is 90 kg and the plane debris is 900 kg. The person and the object fall from a height of 12 km. 
 A: You won't survive even if you managed to jump from a platform before hitting the ground.
The answers given by other users state that you need to apply a large force on the floor (or whatever) to save yourself. While it is certainly true that you need to apply a large force (impulse) to stop your motion, it will not save you.
Why is that you can't survive when you fall hit the ground at a high velocity?
The answer lies in the following equation:
$$impulse = \Delta p = Ft$$
To reduce your momentum to zero suddenly, you need a large impulse. The collision times are very small. This would mean a very large force would act on your body for a short interval of time. This would damage the body severely and hence you are likely to die.
Why does jumping off the floor (or a surface) just before falling does not save you?
To stop your motion by jumping off from the surface, you'll need to apply a large force. This large force is as lethal as hitting the ground. This will kill you.
Suppose you jumped off just before hitting the ground. The impulse you'll need to stop yourself is exactly equal to the impulse you would receive if you had hit the ground. Is that any better? Not at all. It does not help you in any way.
Suppose you jumped off the surface somewhere in mid-air. You'll first need an impulse to get yourself off the surface or slow yourself down. You can either take in a large impulse to slow you down too much (which could potentially be lethal) or a short impulse to slow you just a little bit. If you take high impulse but not lethal, that could reduce the net damage you would take when you hit the ground. If you don't slow down much, it serves no purpose. You continue to move at a high velocity and will eventually hit the ground at an even higher velocity.
At best, you can reduce the damage you take when you hit the ground.
Wait! There is a way out! (not sure if this case is allowed by the question as you don't jump)
Use rockets or jetpacks or a collection of platforms.

Shoot each platform downwards every 5 seconds to reduce your velocity. This probably does not count as jumping but this is a good method.
If you haven't realized, shooting platform is as good as using a jetpack. Jetpack spits out some exhaust gasses instead of shooting platforms.
A: The mistake is in not considering relative velocities. Suppose you can jump upwards at a 3 m/s (made up number that is in the right ball park). But you and the object are falling at 50 m/s. When you jump, you are not magically moving upwards at 3 m/s, you are moving downwards at $50 - 3 = 47$ m/s, which is not much of an improvement. 
If you can jump upwards at 50 m/s just before you hit the ground you so be stationary relative to the ground ( and just above the ground) when you jump, and will survive. But if your body can generate and survive that much amount of force and acceleration, then this isn't going to be a fatal speed to hit the ground at anyway. 
Basically, jumping will only help in a very narrow range of velocities where you are falling slightly too fast to survive, and the small difference jumping makes is enough to put you into the severely injured but barely alive category. 
And the details of how you land probably have a lot more of an effect on the outcome than the small reduction in velocity you get from a perfectly timed jump.
A: Short answer - NO.
Terminal velocity for a skydiver is about 200 kph. The upward acceleration caused by the jump would need to be sufficient to cancel out about 160 kph of this (Most people can survive a 40 kph impact with relatively little trauma).
$K.E. = \frac{mv^2}{2}$ so the energy of impact varies as speed squared.
Also, note that the first thing that happens at 12 km is you pass out from oxygen deprivation. Max alt recommended for breathing without a mask is ~3 km.
This is the 'Jumping in a falling elevator' problem in another guise. See Mythbusters for a complete explanation. Myth Busted
A: If you are falling with a piece of debris then conservation of momentum is applicable as external force is absent. Gravitational force is an internal force.
At the moment of jumping, velocity of the centre of mass of system will be unaffected by the fact that you jump off the debris. The acceleration and velocity of the centre of mass will be same before and after the jump because the force that you push the debris with is internal and internal forces do not change the motion of Centre of mass.
If somehow you apply so much force that your velocity becomes zero and the gain in the velocity is such that the momentum is conserved then it is possible to survive a fall.
Applying conservation of momentum,
$mv+MV=m(o)+MV'$
$V'=\frac{mv+MV}{M}$ 
During free fall, velocity is independent of mass,
$V'=\frac{V(m+M)}{M}$
We can find the velocity as the function of Height.
$v^2=u^2+2gH$
And if initial velocity is 0 then,
$v=\sqrt{2gH}$
Therefore, $V'=\frac{\sqrt{2gH}(m+M)}{M}$
If you are able to give the debris this velocity (V') at the height H, then you will survive the fall if you jump at the right height and time because even after jumping earth will pull you. So, if you jump too early then there will be enough height, on the course of which you will gain a lot of velocity before hitting the ground and all that velocity will be lost in a short interval of time, producing enormous impulse. The impulse would be so great that your body won't be able to withstand it.
But physically, you cannot generate that much impulse by jumping on the debris. 
If it were possible to generate that much impulse, even then it would have shattered your bones.
A: It's not the fall that kills you, it's the sudden stop. 
The answer completely depends on the terminal velocity of the person and object (s)he is standing on and also on the athletic ability of the individual. Terminal velocity for a person falling on a parachute is about $5$ meters/second.  If a person's leap achieved a velocity of $5 ms^{-1}$ (s)he would raise her/his center of mass about $1.25$ meters ($4$ feet) comparable to an Olympic high jumper or NBA slam dunk winner. (To a first order, assume the mass of the object is much greater than the mass of the individual.)  
So if the terminal velocity were very slow, and the individual very athletic, and the individual's timing and trajectory very precise, just maybe.  In the real world, no. 
Footnote: The relationship of initial velocity to height achieved is:
$h = \frac{v^2}{2g}$
Derived from equating energy.  
Gravitational potential energy = $mgh$
Kinetic energy = $\frac{mv^2}{2}$
$mgh = \frac{mv^2}{2}$
$h = {v^2}{2g}$
$g \approx 10 ms^{-2}$
A: I just want to put up the point that will a object be able oppose the force greater to oppose the gravitational force because he had to jump with a force of >9702(approx) which is harder to achieve for a normal person without any suits.And also would he be able to stand on the debris without so much of air pressure.   
