# KE vs Momentum as a standard

1. Dec 29, 2008

### bluejacket

Ok, here's the problem. I am evaluating multiple crossbows to predict the changes caused by various arrow weights. The original arrow mass and velocities are available. I input the two formula's: KE = 1/2MV^2 and MO = MV and solve for each, and then use those results with new arrow weights to solve for the new velocities. My problem is that I get different results.

Which should prove to be the most accurate predictor and why?

I know the results will not be pure because they don't include friction and other variables, but just based on the formula's above I was expecting closer results.

2. Dec 30, 2008

### bluejacket

Let me ask this a different way.
I know the current speed and arrow weights for several models.
Using basic mathmatical priciples I would assume I could use that info to calculate either the Kinetic energy (KE = 1.2mv^2) , or the momentum (MO = mv).
I would further assume that having solved for those values I could then use new arrow weights to back track and find the new resulting velocities.
I would think that if I use the same variables in the 2 formulas I should get the same new velocities. I do not????
Example: Original arrow mass = 425gr Original velocity = 405fps
New arrow mass = 505gr New velocity = ???

My results using the KE formula gave a new velocity of 372fps.
" " " " MO formula gave a new velocity of 341fps.

What am I missing?

3. Dec 30, 2008

### A.T.

I would assume the kinetic energy to be constant for one crossbow and different arrow types.

A crossbow applies force over a distance. I assume that this distance doesn't depend on the arrow type. And the force only depends in the current position of the arrow along this distance. So the integral of force over distance (work) is constant.

I don't quite understand why you expect the momentum to be also constant for one crossbow and different arrow types.

Last edited: Dec 30, 2008
4. Dec 30, 2008

### bluejacket

First off, thanks for the responce. Kinetic Energy was my first standard and I'm glad to see that echo'd as appropriate. It has been a long time since I've thought through this stuff, and I looked at the momentum formula to cross check myself. I know that momentum would go down with increased arrow mass, but I guess I just got hung up in trying to work the respective formula's forward and back with the same variables.
Anyway, I think I see my error, thanks.

5. Dec 31, 2008

### A.T.

If KE is to be the same, then momentum goes up with increased arrow mass. For example: If the arrow mass is 4 times bigger, then it's velocity is only halved. Therefore the momentum doubles.

When you look at the formulas, you'll see that you cannot have two moving bodies with different masses but same kinetic energy and momentum.

Last edited: Dec 31, 2008