Calculating the drop of a fleeing deer

[AvidArcher]

As my name indicates, I enjoy archery (specifically bowhunting). I am in the process of determining the ideal weight (measured in grains) for a set of arrows that I am making. I want to have the heaviest arrow possible (i.e. for optimal momentum and penetration), but I want to still have requisite arrow velocity to achieve impact before the deer has an opportunity to move out of the arrow's path.

Typically when a deer hears an unfamiliar noise (e.g. a bow being fired), it's natural reaction is to crouch (dropping low to the ground) and then spring forward and away from the noise--all in one fluid motion. I am trying to calculate the time that it takes for a deer to drop 2.5 inches. 2.5 inches represents the radius of a deer's vital zone. Let's assume that an average deer weighs 150lbs.

Any ideas on how to come up with this calculation?

I will be sure to share my full analysis with everyone after it is completed. Thanks in advance!

 

[negitron]

It can't drop faster than gravity. So, there's your upper bound.

 

[turbo]

I wouldn't bother as much about arrow-weight and drop, if you can more effectively silence the twang of the bow-string. You can also make improvements in getting the deer closer to you and UNDER you, to reduce its ability to evade.

As for arrow weight, the weight is not always as important as the composition of the shaft. Many years back, I was tinkering with alloy and fiberglass shafts and an old bow-hunter schooled me. He invited me into his shop, and showed me some arrows with various shafts. We then dropped the arrows straight down from shoulder-height into pine boards. Guess which ones penetrated and stuck? The old-fashioned cedar-shafted arrows.

 

[berkeman]

As my name indicates, I enjoy archery (specifically bowhunting). I am in the process of determining the ideal weight (measured in grains) for a set of arrows that I am making. I want to have the heaviest arrow possible (i.e. for optimal momentum and penetration), but I want to still have requisite arrow velocity to achieve impact before the deer has an opportunity to move out of the arrow's path.

Typically when a deer hears an unfamiliar noise (e.g. a bow being fired), it's natural reaction is to crouch (dropping low to the ground) and then spring forward and away from the noise--all in one fluid motion. I am trying to calculate the time that it takes for a deer to drop 2.5 inches. 2.5 inches represents the radius of a deer's vital zone. Let's assume that an average deer weighs 150lbs.

Any ideas on how to come up with this calculation?

I will be sure to share my full analysis with everyone after it is completed. Thanks in advance!

Welcome to the PF. The deer can't crouch faster than it would free-fall, so that's a good approximation.

To calculate how long it takes something to fall 2.5 inches (0.064 meters) from release, you use this kinematic equation of motion:

y = 1/2 g * t^2 (simplified a bit for your situation)

g = 9.8m/s^2 (the acceleration due to gravity)

So 0.064m = 0.5 * 9.8m/s^2 * t^2

which gives t = 0.11 seconds

(somebody please double-check my math)

 

[negitron]

Looks good to me, berk.

 

[AvidArcher]

Welcome to the PF. The deer can't crouch faster than it would free-fall, so that's a good approximation.

To calculate how long it takes something to fall 2.5 inches (0.064 meters) from release, you use this kinematic equation of motion:

y = 1/2 g * t^2 (simplified a bit for your situation)

g = 9.8m/s^2 (the acceleration due to gravity)

So 0.064m = 0.5 * 9.8m/s^2 * t^2

which gives t = 0.11 seconds

(somebody please double-check my math)

Interesting. I was under the impression that the mass of the deer would come into play. It seems to be left out in the above calculation. Isn't it true that as an objects mass increases its rate of acceleration decreases? I know I'm splitting hairs here...

 

[berkeman]

Interesting. I was under the impression that the mass of the deer would come into play. It seems to be left out in the above calculation. Isn't it true that as an objects mass increases its rate of acceleration decreases? I know I'm splitting hairs here...

Nope. The acceleration due to gravity "g" is independent of mass. It's that old "drop a feather and a rock on the moon" demonstration thing. Quiz Question -- why did they do that on the moon?

 

[berkeman]

Now, that having been said (and is true for the deer drop example), the heavier your arrows, the less of their forward velocity they will lose due to air resistance. So a lighter arrow has a higher velocity off of the bow and shoots flatter as a result, but a heavier arrow will carry more of its energy to the target.

I vote for the flat-shooting option, though. Especially with a bow -- like you say, too much goes on while the arrow is in the air, so it's best to minimize flight time, IMO.

 

[AvidArcher]

Now, that having been said (and is true for the deer drop example), the heavier your arrows, the less of their forward velocity they will lose due to air resistance. So a lighter arrow has a higher velocity off of the bow and shoots flatter as a result, but a heavier arrow will carry more of its energy to the target.

I vote for the flat-shooting option, though. Especially with a bow -- like you say, too much goes on while the arrow is in the air, so it's best to minimize flight time, IMO.

Yes this is the conundrum that faces all bowhunters--the balance between velocity and momentum. With a razor-sharp broadhead, relatively little kinetic energy is needed to penetrate a deer's ribcage and vital zone. Still, some errant shots can venture towards the shoulder region which can be a tough cookie to crack without ample arrow momentum. An arrow with a high mass will penetrate through the shoulder bone and into the lungs. Of course, you have to hit what you are aiming at to achieve penetration--hence the need for balance!

 

[AvidArcher]

OK gang...here's some rough numbers.

My maximum effective range is 36.58 meters (distance the arrow travels).
Speed of sound is 332.23 m/s.

Using these two numbers, I can estimate that it will take the deer 0.11 seconds to hear the bow.

Studies have shown that a deer's reaction time is approximately .2 seconds.

As berkeman wisely calculated, it will take the deer .11 seconds to drop to the edge of the kill zone.

Add these times up, and we get 0.42 seconds from the time of the shot until impact.

This would require an average arrow velocity of 87.07 meters/sec. I have shot my bow through a chronograph, and I know that it is capable of achieving this type of speed. Still, this speed is only measured as soon as the arrow leaves the bow. How can I account for a coefficient of air friction and find my average velocity for the arrow as it travels the 36.58 meters from the bow to the deer?

Thanks again to everyone for their help!

 

[negitron]

How far away can you reliably shoot through your chronograph?

 

[russ_watters]

This would require an average arrow velocity of 87.07 meters/sec. I have shot my bow through a chronograph, and I know that it is capable of achieving this type of speed. Still, this speed is only measured as soon as the arrow leaves the bow. How can I account for a coefficient of air friction and find my average velocity for the arrow as it travels the 36.58 meters from the bow to the deer?

The drop in speed due to drag is negligible over such short distances.

 

[berkeman]

How far away can you reliably shoot through your chronograph?

When I was actively bowhunting, I reliably could hit an 8.5x11" piece of paper (in landscape orientation) at 40 yards, which is about what the OP is calling his max effective range. That's probably about the size of his chronograph opening, so yeah, that would be a great experiment for him to try. Initial velocity versus final velocity at max comfortable range.

 

[berkeman]

BTW, just from memory, I'd guesstimate that the velocity at 40 yards looked to be about 80% of the initial velocity. But definitely it would be best to measure it.

 

[sganesh88]

Poor deer. A group of wicked humans planning to kill it using math! :D

 

[AvidArcher]

ha! Survival of the fittest.

I will shoot my current arrow setup through the chronograph at point blank and 10 meters out. This should give me an initial and final velocity to calculate the rate of arrow deceleration.

Look for pics later!

 

[AvidArcher]

Ok, measured initial velocity today: 282 feet per second

Hoping to get final velocity recorded tomorrow. Stay tuned!

 

[Cyrus]

This calls for a good ole experimentation. Good hunting.

 

[berkeman]

ha! Survival of the fittest.

I will shoot my current arrow setup through the chronograph at point blank and 10 meters out. This should give me an initial and final velocity to calculate the rate of arrow deceleration.

Look for pics later!

You might want to get at least 10, 20, and 30 yards, and several shots at each, to get a better data set. Definitely post some pictures!

 

[russ_watters]

Guns don't kill people, math kills people!

BTW, just from memory, I'd guesstimate that the velocity at 40 yards looked to be about 80% of the initial velocity. But definitely it would be best to measure it.

That's a lot less than I would have expected, but when you average it out, you're talking about the average speed still being more than 90% of initial.

And yeah, as an engineer, if you can get a measurement, it's always preferable to speculating.

 

[berkeman]

I could certainly be overestimating the slow-down, so it will be interesting to see what the data say. The fletching at the back of the arrow provides most of the drag, it seems.