Derive a general formula for finding range with a spring launcher

[Bobbert]

Homework Statement

So I made a spring launcher that will fire marbles. I need to derive a general formula for finding range for any given angle and x value.

Homework Equations

1) v₂²=v₁²+2ad
2) E_k=E_s
3) 1/2mv²=1/2kx²
4) d=v*t
This projectile was shot from 1.1m off of the ground.

The Attempt at a Solution

I have no idea where to start.

 

[Cyosis]

This is an experiment you're doing at home? The equations you've noted down do not take resistances into account. Marbles are relatively heavy though so you should be alright with these equations to some extend.

Using equation 3) you can calculate the velocity with which the marble leaves the spring, you can then split v up in its x and y components. When it leaves the spring it's trajectory will roughly be a parabola. The x-distance is given by your equation 4. The equation that is missing is the one that describes the motion in the y-direction. Can you come up with an equation for the y-direction.

hint: what is the only force on the object and in which direction does it point?

 

[Bobbert]

Yes it is an experiment I am doing at home and I am not too worried about resistance.
Ok, I am going to assume it is
v₂=v₁+at because I need time, and I don't have distance.
t=v₁sinθ/a because v2 is 0 halfway up
so
t=2(vsinθ/a) still am not accounting for the 1.1m off the ground (not sure how to do that)

1/2mv²=1/2kx²
v=((kx²)/m)^1/2 but I only want the x speed so
v=(((kx²)/m)^1/2)cosθ

So
d=v*t
d= (((kx²)/m)^1/2)cosθ * 2(vsinθ/a)

Still does not accounting for the 1.1m off the ground

 

[rock.freak667]

For you to account for the 1.1m you will need to use the formula y=y₀+u_yt-1/2gt²

then to get the time for the entire motion, set y=0 and use the quadratic equation formula for t.

 

[Cyosis]

To clear somethings up what is this range you want to find?

t=v1sinθ/a because v2 is 0 halfway up

Not true since you start at 1.1 meters and the projectile lands at 0 meters. The general kinematic equation you're looking for is $s=s_0+v_0t+\frac{1}{2}a t^2$.

Edit:kept the post open for way too long, not adding anything to rock's explanation.

 

[Bobbert]

I am looking for horizontal range.
My equation sheet has the formula
d_y=v₂t-1/2at²

what does y₀/s₀ stand for?

 

[rock.freak667]

I am looking for horizontal range.
My equation sheet has the formula
d_y=v₂t-1/2at²

what does y₀/s₀ stand for?

displacement.

There should also be a t in the formula for d_y.

BUT your equation assumes you started with 0 distance (vertically since you said d_y)

 

[Bobbert]

Sorry about the t.

if s₀ stands for displacement (it should be -1.1 then?), then I set s = 0 and do the quadratic?

EDIT: Also why is yours +1/2at^2 where as mine and cyosis is -1/2at^2 ?

 

[rock.freak667]

Sorry about the t.

if s₀ stands for displacement (it should be -1.1 then?), then I set s = 0 and do the quadratic?

well it can be -1.1 but we took the origin to be where the marble will land, 1.1m below where it was launched.

EDIT: Also why is yours +1/2at^2 where as mine and cyosis is -1/2at^2 ?

well the general form is

s=s₀+ut+1/2at²

Cyosis put a - sign because the acceleration is due to gravity which acts downwards, which is usually taken as -ve.

 

[Bobbert]

Ok so I did:
d_y=v_1yt+1/2a_yt²
-1.1=v_1yt-4.905t² 9.81 for g here.
4.905t²-v_1yt-1.1

t=-v_1y+/-(((-v_1y)²-4(4.905)(-1.1))^.5)/2(4.905)
t=-vsinθ+(((-vsinθ)²+21528)^.5)/9.81 (has to be positive or time will be negative)

then from before:
1/2mv²=1/2kx²
v=((kx²)/m)^1/2 but I only want the x speed so
v_x=(((kx²)/m)^1/2)cosθ

now put those both in d_x=v_x*t

d_x=(((kx²)/m)^1/2)cosθ * (-vsinθ+(((vsinθ)²+21.582)^.5)/9.81)

d_x=(((kx²)/m)^1/2)cosθ * (-(((kx²)/m)^1/2)sinθ+(((((kx²)/m)sinθ)+21.582)^.5)/9.81)

not sure if I can simplify further or if I made a mistake.

 

[rock.freak667]

You should get

$$t= \frac{vsin\theta+\sqrt{v^2 sin^2 \theta-4(\frac{1}{2}g)(-1.1)}}{g}$$


then put that into d_x=(vcosθ)t


(sin2θ=2sinθcosθ)

 

[Bobbert]

Can you show all your work on how you got t, because I keep getting a negative in front.

also fixed my final t line.

 

[rock.freak667]

Can you show all your work on how you got t, because I keep getting a negative in front.

also fixed my final t line.

$$-1.1=(vsin\theta)t-\frac{1}{2}gt^2$$

$$\Rightarrow \frac{1}{2}gt^2-(vsin\theta)t-1.1=0$$

$$a= \frac{1}{2}gt^2 \ b= -vsin\theta \ c=-1.1$$

$$t = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

 

[Bobbert]

Thanks
I am going to assume here g = 9.81 not -9.81 or else the inside of the root could fail.