Understanding Physics of Arrow Flight

[MALON]

My dad gave me this question:

"My most recent bout with physics was in my article on arrow flight. I have a question for you.
I was told many years ago that an object (ball, arrow etc.) shot or thrown, (picture a pitcher or archer) does indeed continue to accelerate (increase in speed) for a short time after the force is removed before the forces acting against it begin its deceleration and drop. Is this correct?
So if this is incorrect then an arrow shot from a bow, immediately begins its deceleration the instant it leaves the string. "

I told him that in general physics, once a force is removed from something in this type of case, it has no reason to continue its velocity increase, so it slows down. I used an archery analogy for him: Picture a full length drawnback bow, which we will call 0. Now picture bow at rest, we will call that 1. I told him that the arrow will start slowing down once the string reaches 1, because the string has no more forward pushing forces acting on it, so it can't continue to increase the velocity of the arrow.

I also did tell him that I think that that maximum point of speed is somewhere around 3/4 and that the arrow will start slowing down after that point. I don't actually know if that's correct, it just seems somehow correct to me. And for all your archery buffs, I am talking about a recurve bow, not a compound. I told that a compound probably has maximum speed much closer to 1.

The only explanation I could give him as to why it may continue to accelerate at all would be a wave effect of compressions and rarefactions, like the way when you throw a water balloon, it jiggles and wiggles in the air, jutting it forward sometimes. Same principle that when you are increasing the velocity the arrow (or any object) the atoms are compressed and upon being released they are then then rarified and then(? ha) cause a SLIGHT increase in velocity for only a moment, and only visible at the sub-atomic level.

Thanks guys!

 

[MALON]

Hm...now that I've been thinking...I think that the arrow does not start to decrease velocity at 3/4, but much closer to 1.

 

[Danger]

Because of the 'springiness' of the bow, I suspect that there is still some acceleration after the resting position, but certainly not after the arrow and the string have parted company. My analogy for that is a bullet continuing to accelerate briefly after leaving the muzzle of a gun because there is still an overpressure zone behind it until the gases dissipate.

 

[DyslexicHobo]

A good analogy for this is a spring.

Lets go through this step by step:
-The arrow is cocked back and ready to fly
-The arrow is released
-The string and the arrow accelerate at the same rate
-The string and arrow reach the rest position at the same time
-Due to Newton's first law, the arrow continues to move forward at constant velocity (the velocity it was traveling at x=0, assuming 0 is rest), and the string begins to decelerate and go back to the rest position (due to tension).

The only reason I can see for the arrow still being accelerated past the bow's rest position is that air resistance is slowing the arrow down during the string's acceleration, such that after the rest position, then string is still moving faster than the bow. I couldn't imagine this having ANY effect on any types of calculation, though, because the amount of acceleration would barely be calculable.

I suppose your theory about waves in the air could work... there is obviously some sort of waves in the air when a bow is shot, because it makes such a cool sound!Do we all agree that, in a perfect environment, the arrow is not accelerated after is passes through x=0?

 

[Danger]

You're probably right, but I'd like to address the point that I brought up just in case.
What I was thinking is that the bow 'whips' past its resting position. It of course begins decelerating at x=0, but at that point is it not still causing the string to accelerate? If so, then the arrow is also accelerating.
I do totally agree, however, that the effect is negligible.

 

[DyslexicHobo]

Well I thought of the string as a spring. You can imagine that the middle of the string on the bow is a spring, and you're simply compressing the arrow on top of the spring.

A spring's maximum velocity is at x=0, and the force's vector also changes direction at x=0. This is, however, only in an ideal environment. You very well could be right, due to some conditions that I can't think of.

 

[Danger]

Maybe the difficulty here is in my approach. It's actually the bow, not the string, that provides almost all of the propulsion. There is some minor elasticity to the string, however, so I envision it still 'catching up' to the bow after the latter has passes x=0. While the bow tips will be decelerating after that point, the elastic rebound of the string might cause it to accelerate a fraction longer.
I really don't know, though, and certainly can't back it up with numbers. It's just something that feels right.

 

[DyslexicHobo]

That's a good point, Danger. It appears that I was envisioning it incorrectly. You are correct that the bow will continue to "uncurve" past the point of x=0. Because of this, it seems that the bow will pull on the rope, making it accelerate a very minimal amount past x=0 (this is limited by the inelasticity of the string).

 

[DaveC426913]

Are we not missing the point here? Surely the OP's OP uses the bow and arrow as merely an example (and a poor one at that, since it complicates the issue with the bowstring).

Surely the crux of the OP's question is "...an object (ball, arrow etc.) shot or thrown, (picture a pitcher or archer) does indeed continue to accelerate (increase in speed) for a short time after the force is removed before the forces acting against it begin its deceleration and drop..."

And the answer is: absolutely not.

Once the mechanism applying the force (be it the bow or the pitcher's arm, or the gun that fires the bullet) stops applying it, the object begins to slow down.



In fact, even this is not telling the real story. The forces that slow down the arrow/ball/bullet are actually acting even as the bow/arm/gun is accelerating the projectile. The retarding forces begin the moment the projectile begins accelerating.

 

[Danger]

You're absolutely right, Dave. I got so caught up in the archery analogy that I actually forgot all about the original part of the question.