Energy conservation and projectile motion

[Dark_Rak3r]

Homework Statement

A spring is pulled back and launches a ball. The ball flies into the air at a perfectly horizontal angle. How far will the ball go before it hits the ground?

Variables for the FINAL equation are:
∆x - distance the spring is pulled back
g - acceleration of gravity
m - mass of the ball
k - spring constant
∆R - range the ball will fly
h - height above the ground at the point that the ball passes the end of the launcher

Homework Equations

1/2mv^2 - kinetic energy
1/2kx^2 - elastic potential energy

v=√(kx^2/m) - velocity derived from the kinetic energy equation

The Attempt at a Solution

I broke it up into 2 separate parts, one dealing with conservation of energy and the spring, and the other dealing with kinematics and the projectile motion of the ball in the air. I managed to find the velocity of the ball when it leaves the launcher, but beyond that I really had no idea where to go, as I need to find some way to eliminate both the velocity and time variables from the final equation, which would be much easier to use if I was able to.

 

[rock.freak667]

Just consider vertical motion to get the time parameter and then just put that into the equation for horizontal motion

 

[Dark_Rak3r]

I would, but I need to somehow come up with a final equation using only those variables.

 

[rock.freak667]

I would, but I need to somehow come up with a final equation using only those variables.

If R is the horizontal distance traveled, then $R=vcos\theta t$

Consider vertical motion now, use the fact that when the ball hits the ground, the vertical displacement is zero (this will get you the time parameter).

 

[Dark_Rak3r]

Why would the vertical displacement be zero? The ball is assumed to be starting above the ground, in which case the vertical displacement would be a non-zero number (again, the ball is being shot at a perfectly flat angle, only completing one half of a parabola). Your math is also kind of going right over my head, is there a simpler explanation for what you mean? Also, what would I do with the time? I could solve this easily enough if it was an allowed variable, but it can't be used in the final equation for the problem.

 

[rock.freak667]

Why would the vertical displacement be zero? The ball is assumed to be starting above the ground, in which case the vertical displacement would be a non-zero number (again, the ball is being shot at a perfectly flat angle, only completing one half of a parabola). Your math is also kind of going right over my head, is there a simpler explanation for what you mean? Also, what would I do with the time? I could solve this easily enough if it was an allowed variable, but it can't be used in the final equation for the problem.

Let's start over now that I've read the horizontal angle thing.

You correctly got the equation for the velocity,v.

We know that ∆R=vt, but you don't have t.

We also have the equation s=s₀+Vt-(1/2)gt² for vertical displacement.

The initial vertical displacement is h. When the ball hits the ground, won't it have a vertical displacement of zero? (When the ball is touching the ground, you would not say it has a vertical displacement).

So you can now find t in terms of the parameters that you have. When you get that, put that into the formula for ∆R, and you will get the equation you need.

 

[Dark_Rak3r]

I see what you mean by the displacement being 0. What does s represent in the equation you gave? The sine of the launch angle?

 

[rock.freak667]

I see what you mean by the displacement being 0. What does s represent in the equation you gave? The sine of the launch angle?

No no, s is just the vertical displacement and s₀ is the initial vertical displacement

 

[Dark_Rak3r]

So, just to keep the variables similar, the equation could be written as:
h=h_i+vt-(1/2)gt²
where all variables correspond to those in my first post?
Also, if h is the vertical displacement, can I use:
∆d=v_i∆t+1/2a(∆t)²

where v_i∆t=0 because the initial vertical velocity is 0, and the rest can be solved as ∆t=√(2∆h/a)

 

[rock.freak667]

So, just to keep the variables similar, the equation could be written as:
h=h_i+vt-(1/2)gt²
where all variables correspond to those in my first post?

Here is where you need to be careful, if you take the origin to be where the ball is launched, the initial vertical displacement is zero and you want to put s=-h.

OR otherwise, you want s=0 and s₀=h.

and the v in this equation is the initial vertical velocity which is zero. Can you get t now?

 

[Dark_Rak3r]

So t=√(2h/g) and the final equation is ∆R=√(kx²/m)*√(2h/g)

 

[rock.freak667]

So t=√(2h/g) and the final equation is ∆R=√(kx²/m)*√(2h/g)

Yes and you have all of the variables you needed in this equation.

 

[Dark_Rak3r]

Thanks for the help. I'll definitely come back to these boards in the future if I need anything else.

EDIT: AAAARRRRRRGGGGGGG! I just realized that I had figured out the time equation earlier without realizing it was the right equation. I took the equation h=v_i∆t+(1/2)g(t)² and figured out that v_i∆t=0 and that ∆d=(1/2)g(t)² could become ∆t=√(2h/g). While I'm not sure that that method will work all the time it is aggravating to know that I would have still gotten the right answer even if my explanation was incorrect.