launching spring at targets (physics)?

Question

I have a question regarding physics.
Its about launching a spring at various targets (bucket) at different distances and heights.

WHat would be the equation to find the amount of stretch from equilibrium (in the string) i.e. x. We are given the vertical height, horizontal distance, the constant of the spring (k) and

Any help would be appreciated. Thank you in advance

Answer 1

You also need the angle of launch.

The spring has mass m.
Assuming ideal behavior (do not let the coils of the spring touch), the spring will store energy equal to
.5*k*x^2
where x is the stretch from equilibrium.
This relates to initial speed, v as
.5*k*x^2=.5*m*v^2

now, the range will be
v*cos(th)*t

to find t, use the launch height, h, in the equation
y(t)=v*sin(th)*t-.5*g*t^2

when y(t) = the height of the target, you have impact
solve for t in terms of v and plug into the range, and then relate v to x.

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