In the reality television show "Amazing Race," a contestant is firing 12 kg watermelons from a slingshot to hit targets down the field. The slingshot is stretched from its equilibrium length by a distance of 1.4 m, and the watermelon is at ground level, 0.5 m below the launch point. The targets are at ground level 15 m horizontally away from the launch point. Calculate the spring constant of the slingshot (in N/m). (Assume the angle that the watermelon's velocity makes with the horizontal at the launch point is the same as the angle the slingshot makes with the horizontal when pulled back. Also assume the equilibrium length of the slingshot is negligible.) slader

k = 930 N / m

Explanation:

For this problem we will solve it in parts, let's start with the conservation of mechanical energy

Starting point. Lower

          Em₀ = + U₁

Final point. Higher

         = U₂

as there is no friction, energy is conserved

          Em₀ = Em_{f}

          ½ m v² + mg y₁ = m g y₂

where y₁ is the initial height of y1 = -0.5 m and y² the final height y² = 15 m

Let's find speed when getting out of the sling

         v = √ (2g (y₂-y₁))

let's calculate

         v = √[2 9.8 (15 - (-0.5))]

         v = 17.43 m / s

Now we can use Newton's second law.

The force applied by the sling is in the direction of movement (inclined) and the weight is in the vertical direction.

X axis

         Fₓ = m aₓ

in the problem they indicate that the direction of the velocity at the end of the sling is the same direction of the force,  

         F_{e} cos θ = m a cos θ

let's replace the elastic force

        k Δx = m a

Y axis

        F_{y} - W = m a_{y}

         k Δx sin θ - m g = m a sin θ

let's write

              k Δx = m a        (1)

            k Δx sin θ - m g = m a sin θ

Now let's use kinematics to find the acceleration in the sling, the direction of these accelerations ta in the direction of elongation

           v² = v₀² +2 a Δx

as the system starts from rest v₀ = 0

           a = v² / 2Δx

           a = 17.43² / (2  1.4)

           a = 108.5 m / s²

we substitute in equation     1

            k = m a / Δx

            k = 12 108.5 / 1.4

            k = 930 N / m

see earlier answer