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. Show your work. Write your answers in terms of k, L. M and g. Check units/dimensions for each answer. don't forget gravity and place your origin at the uncompressed spring position.

Explanation / Answer

A) PE of spring will be convert in to KE of mass

=> (k*L^2)/2 = (M*v^2)/2

=> v = L(k/M)^0.5 m/s , v is speed of the ball when mass leaves the spring

B) v^2 = u^2 + 2*g*h

=> 0 = k*L^2/M + 2*(-g)*h

=> h = k*L^2/(2Mg)

so maximum height above the uncompressed position ={[ k*L^2/(2Mg) ] - L } m   

C) for just leave the spring

kinetic energy should be just enough to reach the ball at height = L

=> v^2 = u^2 + 2*a*S

=> 0 = u^2 +2*(-g)*L

=> U = (2gL)^0.5

now KE = PE

(M*U^2)/2 = (k*L^2)/2

=> minimum spring constant k = M(U/L)^2 = 2Mg/L

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