A bow string pushes an arrow say at 50N force for 0.05 seconds.

 

If the arrow is made of lead, it will not reach very far.

If the arrow is made of balsa wood, it will not reach very far.

 

If the arrow has certain mass, will 'match' and will reach the farthest.

 

How to calculate the optimal matching mass for an arrow in order to reach the farthest; for a given force / impetus ?

I think your premise is flawed; the amount of time the force can be exerted depends on the arrow you use. A larger mass arrow will have the force exerted for longer, but with a smaller acceleration; the result will be that the final speed is smaller.

A bow string pushes an arrow say at 50N force for 0.05 seconds.

 

If the arrow is made of lead, it will not reach very far.

If the arrow is made of balsa wood, it will not reach very far.

 

If the arrow has certain mass, will 'match' and will reach the farthest.

 

How to calculate the optimal matching mass for an arrow in order to reach the farthest; for a given force / impetus ?

 

You are given the momentum.

 

You want the smallest mass possible to achieve the highest velocity and gain the maximum energy.

 

The higher velocity the higher the drag force in air, generally proportional to the square of the velocity.

 

So your optimum will depend on the coefficient of drag and frontal area, and coefficient of skin friction and surface area, of the arrow.

Thanks, gentlemen.

The bow and arrow was chosen for the question instead of launching a ball by arm force because a force could be kind of measured to the string tension applied.

 

About the frontal area, it is for a question like this, to consider all arrows identical in shape.

J.C. indicates the smallest mass will achieve the highest velocity. I believe that; but what about the travel distance ?

 

Back to the arm and ball throwing, A paper ball, a lead ball and a baseball -made all same size and aerodynamics- I can assure the baseball will fly much farther.

 

Not the lightest nor the heaviest will reach farther.

 

It will be the one that 'matches' the arm strenght. Now I would like to see that translated into calculations, with bow and arrow instead of arm and ball.

Thanks, gentlemen.

The bow and arrow was chosen for the question instead of launching a ball by arm force because a force could be kind of measured to the string tension applied.

 

About the frontal area, it is for a question like this, to consider all arrows identical in shape.

J.C. indicates the smallest mass will achieve the highest velocity. I believe that; but what about the travel distance ?

 

Back to the arm and ball throwing, A paper ball, a lead ball and a baseball -made all same size and aerodynamics- I can assure the baseball will fly much farther.

 

Not the lightest nor the heaviest will reach farther.

 

It will be the one that 'matches' the arm strenght. Now I would like to see that translated into calculations, with bow and arrow instead of arm and ball.

 

That would be a lot safer. It would keep your arm from heading off into orbit by choosing too light a paper ball.

 

Edit: Seriously, neither arm nor bow is anything close to an idealized "momentum transferring device", as suggested in your OP, that could apply a force of 50N for .05 seconds regardless of how light the arrow or ball.

Edited May 18, 201213 yr by J.C.MacSwell