Design a spring floor system such that the gymnast can achieve maximum height. Y

Question

Design a spring floor system such that the gymnast can achieve maximum height. Your task is to determine a mattress floor with a suitable coefficient of restitution e and a spring system with spring constant k. To test the spring floor system, a 5 kg ball is dropped from a height of y = 3 m. The mass of the spring floor system is 10 kg.

a) Determine the velocity of the sphere just before impact.

b) Determine the final height h reached by the sphere after the first bounce in terms of the coefficient of restitution e.

c) If the maximum additional allowable compression is 40 mm, determine an expression for the optimal spring constant k in terms of e only.

Explanation / Answer

a) From conservation of energy, mgy= 1/2 mv2

So, v before impact= (2gy)1/2

v= 7.66 m/s.

b) K.Ei × e2 = K.Ef

Kinetic energy before collision is 1/2 m (7.66)2= 146.68 J.

K.Ef= mgh from conservation of energy.

So, 146.68e2= mgh

h= 2.99 e2 m.

c) v after impact= v before impact × e

So, va= vb × e

And now, va= 7.66 × e

And, 1/2 mva2 = 1/2 kx2

k= 5 ×(7.66e)2/ (0.04)2

k= 183361.25 e2 N/m2

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