# Physics back of napkin

A project log for Stickvise Crossbow

Every electronics bench should have a crossbow. First crossbow project on hackaday.io

So the springs on Stickvise seem fairly strong, why does my velocity suck? I intentionally didn't do any analysis ahead of time because I just wanted to focus on the idea, not the results. Here I will do a quick grade school physics analysis to see why the pencil didn't go too fast or far.

First let's figure out what we will need to know to calculate projectile velocity:

1. How stiff is the spring (what is its spring constant, k)?

2. How much mass does the spring have to move (sum of moving vise parts mass + projectile mass)?

3. What is the travel (how far does the spring move the mass)?

To find a spring constant for my springs, I simply put a stickvise on a scale, tared it, then compressed the spring loaded jaw by a measurable amount and took a reading from the scale. Here's what I came up with:

1. Spring Constant Calc.

Spring compression: .214" (.00544 meters)

Scale reading: 1500 g (1.5 kg)

Note that the scale thinks it is weighing a mass in earth's gravitational field, so to get the actual spring force from the mass reading, I multiplied the mass (1.5 kg) by 9.81 m/s^2 to get a spring force of 14.715 N.

To get spring constant use Hooke's law for spring force (a good approximation I hope):

and solve for k:

2. Mass the spring has to move

No calculation needed here, I just weighed the parts. It was 55 g for the vise parts, the pencil (not shown) was 3 g for a total of 58g (.058 kg)

3. Spring travel

For this I just measured the jaw spacing before and after cocking the trigger. The total travel is .373" or 0.00948 meters.

Calculating final velocity

Neglecting wind drag and friction, you can find the projectile velocity using conservation of energy equation shown below. The left side is the potential energy stored in the spring, the right side is the kinetic energy put into the moving parts.

solving for velocity you get

hmmm this is 4.57 miles per hour, not breaching the speed of sound any time soon.

When shot horizontally from a height of 1 m, the projectile will travel .926 meters before hitting the ground...

Well one interesting thing I got out of the calculations is that I am better off finding ways to increase the travel of the crossbow (x) rather than I am going to beefier springs or lighter weight on the moving parts. I guess I should probably do all three though because I am so far from an awesome crossbow right now.

Let me know if my calcs are off, the final results were close to what I saw in reality so I'm thinking they are probably correct.

## Discussions 