# Find spring constant, unknown projectile mass, vertical spring gun.

What I have given is:
A projectile with an unknown mass is launched 90 degrees vertically to a height of 0.650 m with a spring deformation of x = 0.031 m. The initial speed of the projectile is v0=3.6 m/sec and the projectile is resting on top of the spring gun "holder" before launch. Find the spring constant k.

## Homework Equations

W = d*F, P = F*v*cos(θ), W = Epot, ay=-g, P=ΔE/Δt, Etotal=Ekin+Epot.

## The Attempt at a Solution

I don't know exactly how to handle this problem, but I've been trying to attack it from many many angles with W = d*F, P = F*v*cos(θ), W = Epot, ay=-g and the relationship between distance, velocity, acceleration and time but it all comes down to me missing the mass and the force in every equation with at least two unknowns in every case. The closest I ever got was to realize that W being done on the projectile = Epot that the spring has before launch and wondering if I could use the W=d*F to figure out the F-value so I could figure the rest of the equations out but
P=F*v seemed plausible as well hoping that cos(θ) = 1 in case of the angle being 0 but I gave up on that because it said in my formulae book that I need a particle with a speed in order to use it and the projectile is a static particle after all.
So what do I do?! It's very important that I hand this in as I might not be able to continue my class if I don't and I prefer doing it on my own but I've come to a point spending hours and hours where I simply don't know what to do ... Maybe I'm just stupid and can't see the solution right in front of me but it really seems hard to me ... Please help! It's really urgent!

Oh and I also got a clue from the teacher: Consider energy relations when you solve this problem. And I did and I still have no answer.

PhanthomJay
Homework Helper
Gold Member
What I have given is:
A projectile with an unknown mass is launched 90 degrees vertically to a height of 0.650 m with a spring deformation of x = 0.031 m. The initial speed of the projectile is v0=3.6 m/sec and the projectile is resting on top of the spring gun "holder" before launch. Find the spring constant k.

## Homework Equations

W = d*F, P = F*v*cos(θ), W = Epot, ay=-g, P=ΔE/Δt, Etotal=Ekin+Epot.

## The Attempt at a Solution

I don't know exactly how to handle this problem, but I've been trying to attack it from many many angles with W = d*F, P = F*v*cos(θ), W = Epot, ay=-g and the relationship between distance, velocity, acceleration and time but it all comes down to me missing the mass and the force in every equation with at least two unknowns in every case. The closest I ever got was to realize that W being done on the projectile = Epot that the spring has before launch and wondering if I could use the W=d*F to figure out the F-value so I could figure the rest of the equations out but
P=F*v seemed plausible as well hoping that cos(θ) = 1 in case of the angle being 0 but I gave up on that because it said in my formulae book that I need a particle with a speed in order to use it and the projectile is a static particle after all.
So what do I do?! It's very important that I hand this in as I might not be able to continue my class if I don't and I prefer doing it on my own but I've come to a point spending hours and hours where I simply don't know what to do ... Maybe I'm just stupid and can't see the solution right in front of me but it really seems hard to me ... Please help! It's really urgent!

Oh and I also got a clue from the teacher: Consider energy relations when you solve this problem. And I did and I still have no answer.
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You are listing a bunch of equations, some of which are not correct. You should use energy methods, since using force methods gets a bit tricky because the spring force is not constant. When you used the energy approach, what equations did you use? You have 2 unknowns, so you need 2 equations. Note that the given initial speed is the speed at which the projectile leaves the spring when the spring has returned to its unstretched length. Note also that the projectile has no speed when it is released, and no speed when it reaches its max height.

When using energy approach I know the following:
Epot=mgh before the spring has become unstretched and it describes the energy of the spring i.e. the energy that the spring has a potential to give to the projectile.
The thing is that I do not know the mass of the projectile...

I also know that Epot(of spring before spring is released) = Ekin(of projectile after spring is released).
Ekin=½mv2.

Oh so v0 is the speed of the projectile after the spring has released and gone back to its equilibrium state? That's how I understood this: "Note that the given initial speed is the speed at which the projectile leaves the spring when the spring has returned to its unstretched length."

Let's say that I set up 2 equations:
I: Ekin=½mv2
II: Epot=mgh

I isolate for m in both of them:
I: => m=Ekin/(0.5*v2)
II: => m=Epot/(gh).
But another problem
III: Ekin/(0.5*v2) = Epot/(gh).

In this case there are two unknowns, Epot and Ekin though I know that Epot(before)=Ekin(after) so I could replace Epot with Ekin but then I am stuck as well because I don't know either Ekin or Epot. And of course I know that Ekin=0J at the max height but if I insert that in equation III then I get 0=Epot/(gh) and can I really isolate for Epot in that to find what Epot is? I feel so uncertain about that method.

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PhanthomJay
Homework Helper
Gold Member
You must use Conservation of Energy, that is, initial total energy = final total energy. Do it in 2 stages: first, for the motion when the projectile is first released to when it leaves the spring; and second, for the motion when it leaves the spring to when it reaches its max given height.

First case: ....what is its initial energy and what is its final energy.
Second case: ...what is its initial energy and what is it's final energy.

Note that energy consists of grav pot energy, kin energy, and spring elastic pot energy.
Be careful as to what initial and final mean: In case 1, its its initial speed is 0 and its final speed is 3.6 m/s; in case 2, its initial speed is 3.6 m/s and its final speed is 0. Etc.

Okay thanks I'll look at that and try to figure it out.

PhanthomJay
Homework Helper
Gold Member
Sorry, I didn't think this problem through clearly. Is this problem worded correctly? As I understand the problem, it cannot be solved when both the mass and the spring constant are unknown. As I see it.

Well the teacher said that I had to ask my group about it for the correct answer and I already handed it in the first time stating: "It can't be solved because the mass is missing" but that wasn't the right answer. But to be honest I did write "The mass of the spring and the projectile is unknown and therefore I can't find the spring constant" but that's not true as the mass of the spring is not needed in this case. But still I want to solve this on my own if it's possible.

Here is the exact wording of the assignments regarding the diagonal throw:
A: Demonstrate the equation x=(v^2/g)*sin(2*theta) through showing the connection between the shooting lenght and the elevation angle for the diagonal throw.
B: After this, use the aforementioned connection to determine the initial velocity v0, for the diagonal throw
C: The, under B determined, v0-value now has to be compared with the value that can be determined from the vertical throw. (I assume that the value that the teacher talks about is the v0-value for the vertical shot).
D (here comes the assignment I am asking for help to): Finally the spring constant k for the spring in the cannon has to be determined.

I mean maybe I could also make a linear graph and then determine the gradient (the spring constant) of it but is that possible?

There are 4 more formulations that the teacher writes that I will summarize:
A: Measure connection between throwing width x and elevation angle theta. Make a (x, sin(2theta) diagram.
B: From the graph determine the inital velocity v0 for the diagonal throw
C: The cannon is positioned with its end so that it's vertical. Determine v0 for the vertical throw. V0 for B and C has to be compared.
D: Finally, the spring constant k has to be determined with "energy considerations". Which is why I posted that on the forum.

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