A vertical spring with constant k is compressed a distance L and a ball with mas

Question

A vertical spring with constant k is compressed a distance L and a ball with mass M is set on top of it. When the spring is released, the ball is fired upward. Find the speed v of the ball when it leaves the spring. Find the maximum height h above the uncompressed position of the spring reached by the ball. If the spring constant is too small, the ball never leaves the spring. Find the minimum value of the constant k for the ball to leave the spring. Write your answers in terms of k, Z, M, and g. Check units/dimensions for each answer.

Explanation / Answer

here,

the spring constant is k

a)

let the speed of the vball be v

using conservation of energy

k*l = m*g*l + 0.5 * m * v^2

v = sqrt(2*(k*l - m*g*l)/m)

the speed of the ball is v = sqrt(2*(k*l - m*g*l)/m)

b)

let the maximum height be h

using third equation of motion

v^2 = 2*g*h

h = ( (k*l - m*g*l)/mg)

h = ( l* (k/mg - 1) )

the maximum height reached is ( l* (k/mg - 1) )

c)

for minimum value of k

gravitational force = spring force

m*g = k*x

k = m*g/x

the minimum value of the spring constant k is m*g/x

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