Question

The desperate contestants on a TV survival show are very hungry. The only food they can see is some fruit hanging on a branch high in a tree. Fortunately, they have a spring they can use to launch a rock. The spring constant is 900 N/m, and they can compress the spring a maximum of 45 cm. All the rocks on the island seem to have a mass of 500 g.
a) With what speed does the rock leave the spring?
b) To what height can the rock be launched?

Answer 1

Answer:

a) v = 18.86 m / s, b)  h = 8.85 m

Explanation:

a) For this exercise we can use the conservation of energy relations.

Starting point. Like the compressed spring

          Em₀ = K_e + U = ½ k x² + m g x

the zero of the datum is placed at the point of the uncompressed spring

Final point. With the spring if compress

           Em_f = K = ½ m v²

how energy is conserved

          Em₀ = Em_f

          ½ k x² + m g x = ½ m v²

   

           v² = $\frac{k}{m}$  x² + 2gx

let's reduce the magnitudes to the SI system

          m = 500 g = 0.500 kg

          x = -45 cm = -0.45 m

the negative sign is because the distance in below zero of the reference frame

       

let's calculate

           v² = $\frac{900}{0.500}$  0.45² + 2 9.8 (- 0.45)

           v = √355.68

           v = 18.86 m / s

b) For this part we use the conservation of energy with the same initial point and as an end point at the point where the rock stops

           Em_f = U = m g h

           Em₀ = Em_f

            ½ k x²2 + m g x = m g h

            h = ½  $\frac{k}{g}$   x² + x

let's calculate

           h = $\frac{1}{2} \ \frac{900}{9.8 } \ 0.45^2$ - 0.45

           h = 8.85 m

measured from the point where the spring is uncompressed