Is the poundage of a bow relevant for the path of the arrow if the arrow mass weight ratio stays the same? In archery, we measure the weight of a bow in pounds. It's measured at 28 " from a pivot point + 1,75 ".
There is also an essential measure for the arrow, which is "grain per pound". It has a certain mass weight measured in grains and this is set into relation to the draw weight. To make this question a little less complicated, we just assume a GPP of 9. So, for a 30 # bow, the arrow weights 270 grain.
Considering that the GPP stays the same, does the poundage of a bow even matter for the arrow flight? Will the path of the arrow be more or less the same for a 25 # and a, let's say, 50 # bow?
 A: A constant GPP means that the mass of the arrow is proportional to the force. In other words, for a GPP value of $k$, the mass of the arrow is equal to:
$$m=kP$$
where $P$ is the maximum force exerted by the bow on the arrow (i.e. the poundage).
The potential energy stored by the bow at full draw is equal to the area under the draw curve (i.e. the force integrated over the draw distance), and the draw weight (aka the poundage) is equal to the maximum of the draw curve. The draw curves of different types of bows are substantially different, which means that different types of bows can store different amounts of potential energy for the same draw weight:

As you can see, a compound bow stores more potential energy for the same draw weight. So let's assume that you're comparing two bows of the same type but different poundage, so that the shape of the curve stays basically the same, but it's just scaled up in proportion to the poundage. Then we can say that the potential energy $U$ stored by the bow is related to the poundage  $P$ by:
$$U=bP$$
where $b$ is a constant, namely, the potential energy stored by a hypothetical 1-pound bow at full draw.
When the bow is fired, some of the potential energy of the bowstring is transferred to the arrow, and the arrow gains some kinetic energy, defined as:
$$K=\frac{1}{2}mv^2$$
for speed $v$. The energy transfer is not perfect - for example, some energy goes into pushing air out of the way of the moving bowstring, and some energy is left in the vibrating bowstring and limbs after firing. Let's suppose that the bow transfers energy with some efficiency $\epsilon$, such that:
$$K=\epsilon U$$
Putting everything together, we have that:
$$K=\frac{1}{2}mv^2=\frac{1}{2}kPv^2=\epsilon U=\epsilon bP$$
In other words, we have the speed of the arrow after firing, and its kinetic energy:
$$v=\sqrt{\frac{2\epsilon b}{k}}$$
$$K=P\epsilon b$$
So there are several conclusions you can make here:

*

*Increasing the poundage of the bow with constant GPP increases the kinetic energy of the arrow, if all other conditions are held constant. This generally gives the arrow more penetrating power.


*Increasing the poundage of the bow with constant GPP does not increase the speed of the arrow, if all other conditions are held constant. The speed of the arrow is determined only by the GPP, the efficiency of energy transfer, and the shape of the draw curve.


*The kinetic energy of the arrow is independent of the GPP. It only depends on the poundage, the efficiency of energy transfer, and the shape of the draw curve. Increasing the GPP gives you a slower arrow with more mass and the same kinetic energy.
A: The bow poundage indicate the amount of energy a bow can transfer to the arrow. The higher the energy transfer to the bow the higher the initial speed.
Under the same conditions and given the same initial speed the laws of motion predicts that the arrows will travel the same path.
But in order for a 50 # to have the same initial speed as a 25 #, approximately twice the amount of work must be done on the bow.Provided the bow can support the load it's more or less the same.
A: A 50# bow exerts twice as much force on the arrow as a 25# bow (over the same distance), giving the arrow twice as much kinetic energy and a higher speed.  A higher speed arrow will experience less vertical drop for a given horizontal distance of travel (and will strike with a greater impact).
