Basically I agree with ShemSeger's answer:
But don't take my word for it, see for yourself:
This guy is a real-life Hawkeye, he very matter-of-factly states that
if he can see it, he can hit it with his bow and an arrow. And if you
watch the video, you'll see that it's true, he hits everything,
including a tiny pill flicked up into the air.
However, I want to add a few things.
In the film(s) we see that she uses the bow to hunt. Therefore, she has learned a "hunting style" to shoot.
So, here comes the part where I disagree with ShemSeger. I don't think that Lars Anderson is a good analogy. I want to reference some great shooters which shoot to kill (so, to hunt). It's the style of shooting which would fit best in the film's framework.
Here's a list:
Just comment if I've forgotten someone :)
So, the answer to your question is: "Yes, it's absolutely possible (even to top that)." Just look at a few of their videos. The accuracy is absolutely fascinating. Also, for example Jeff hunts with a 65 pound bow which is pretty much. A bow like this would be absolutely able to shoot an explosive arrowhead towards a plane (if it would really go down etc. is another question :)).
The funny thing is that the actor was trained an Olympic shooting style which differs from the hunting one. I don't want to go to deep into these differences, because it would be a topic for another question.
This video sums it up pretty well:
Jennifer Lawrence was trained by Olympic archer Khatuna Lorig.
From this article:
To prepare for her role as "Hunger Games" heroine Katniss Everdeen,
Lawrence trained with professional archer and Olympian Khatuna Lorig
to learn how to properly shoot with a bow and arrow.
Edit: The thing with the plane
Because it was mentioned in the comments, I'm going to say something about the plane.
First of all a woman is absolutely able to draw a 65 pound due that the power to draw such comes out of the back muscles.
How far can one shoot with a 65 pound bow? That is of course dependent on the arrow, the bow and the draw length of the individual shooter. All these factors are included in the final value: the arrow's speed. So, to shorten this, we assume a speed of
210 fps (feets per second) or
~64 m/s. She has the longest range when she holds the bow to 45°:
R = (v²/g)*sin(2*a)
R = ((64 m/s)² / (9,81 m/s²)) * sin(90°)
R = 417,53 m
Okay, air resistance apply and so we have about 400m: if the plane was flying beneath 400m it would have been possible for her to shoot it.
Also, the speed of the plane shouldn't be that big problem if you consider videos like this:
One is able to hit exactly the point where two pendulums overlap. It's common sense that it's also possible then to find the right moment to hit something big like an airplane.