Need to find the spring constant to achieve Max Velocity

[MikeyDoubleDEE]

Hello All.
I am mentoring a high school student in my area with his class project for school. He has chosen he wants to launch an object (in our case, a softball) into a 5' diameter area. The idea is to build basically an oversized slingshot using an extension spring as the source of energy.

We arbitrarily picked a distance of 50yds (45.7 m) with a launch angle of 30°. Using projectile motion fundamentals we came up with the following:

d=(Vi²/g)*sin2Θ

Plugging in:
d = 45.7m
Vi = ?
Θ = 30°
g = 9.81m/s²Solving for Vi, I got 22m/s (Which will serve as the final Velocity in my forthcoming spring calcs)

So now we migrate into spring fundamentals. I initially thought to use an engineer's best friend F=ma but quickly realized that the acceleration is not constant in the context of springs and that calculus would need to be leveraged for a diminishing a value. So scratch that.

Settling on the law of conservation of energy, I sought to find the Kinetic Energy of the mass (Softball = .195kg) at my final Velocity of 22m/s.

K=1/2(mv²)

Plugging in:
m = .195kg
v = 22m/s

Solving for K, I got 47.19 Joules

This is where I start to doubt myself:
I found this equation in my old College Physics textbook
Vmax=√(k/m) * A
(A standing for displacement, and k standing for spring constant)

Plugging in:
Vmax = 22m/s
m = .195kg
A = .61m (24" is what we arbitrarily selected as a starting point for our spring selection)

Solving for the spring constant k, I got 2507 N/m (14.315 lbs/in : My mind thinks in pounds and inches, not Newtons and meters)

Is this the correct approach? I don't want to start buying hardware to build our prototype unless I can get some feedback from the physics community that our approach to solving this design problem is valid. Thank you so much for your time!

Disclaimer: I realize that this approach does not take into account gravity during the acceleration of the mass "uphill" at 30 deg. Also, this assumes a frictionless slingshot. And we decided to ignore wind resistance in the projectile motion portion of this problem.

 

[berkeman]

Welcome to the PF.

We arbitrarily picked a distance of 50yds (45.7 m) with a launch angle of 30 yards.

I assume that's a typo, and you mean 30 degrees. Is there a reason you didn't pick the more traditional launch angle of 45 degrees? A low launch angle like that will make it harder to hit the target.

And we decided to ignore wind resistance in the projectile motion portion of this problem.

That's probably not a good assumption. It makes the calculations easier, but you may want to consider making a first order correction to that once you get your initial numbers nailed down.

Also, if you don't impart some spin on the softball, it will act like a knuckle ball in the air, which will impair your accuracy some, IMO. If you can somehow give it some backward spin during the launch, that could improve your accuracy a fair bit.

 

[MikeyDoubleDEE]

Yes that was a typo. I've edited the mistake.

We may recalculate to include wind resistance. That could be a good exercise.

I have thought of the knuckle ball effect and completely agree with you. As we get into prototyping, we will start to troubleshoot some of our findings. It will make for some more meat in his paper that accompanies the project

 

[berkeman]

As a quick reality check on your calculations, you could take a softball out to a local football field and the student could take a video of you throwing the softball the 50 yard distance to hit the center of the field from the goal line. There are several ways that you can analyze the video to see how close your calculation of 22m/s is for Vo. Fun project.

 

[MikeyDoubleDEE]

I like that idea!

 

[berkeman]

Of course, if the student has a better arm, maybe you should shoot the video...

 

[MikeyDoubleDEE]

Can anyone enlighten on my approach to this problem? In particular did I use the correct equation to determine a spring rate with a given Vmax?