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?

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moya1316 moya1316 · Answer 1 · 2021-03-28T12:02:05+02:00

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 = [tex]\frac{k}{m}[/tex] 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 = [tex]\frac{900}{0.500}[/tex] 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 [tex]\frac{k}{g}[/tex] x² + x

let's calculate

h = [tex]\frac{1}{2} \ \frac{900}{9.8 } \ 0.45^2[/tex] - 0.45

h = 8.85 m

measured from the point where the spring is uncompressed