A blob of clay of mass M is propelled upward from a spring that is initially compressed by an amount d. The spring constant is k. What is the ultimate height h above the unstretched spring's end that the clay will reach? A) kd²/(Mg) B) kd²^{/(2Mg) − d} C) kd²^{/(Mg) − d} D) kd²^/(2Mg)

Question

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The potential energy stored in the compressed spring is given by: P = 1/2 k d^2 When the clay is released, this potential energy is converted into kinetic energy: P = 1/2 M v^2 where v is the velocity of the clay when it leaves the spring. The kinetic energy is then converted into gravitational potential energy as the clay rises: P = M g h where h is the maximum height the clay reaches above the unstretched spring's end. Equating these three expressions for P, we get: 1/2 k d^2 = 1/2 M v^2 = M g h Solving for h, we get: h = (1/k) (1/2 M v^2) = (1/k) (1/2 k d^2) = 1/2 M d^2/k At maximum height, the velocity of the clay is zero, so we can find v using conservation of energy: 1/2 M v^2 + 1/2 k d^2 = M g h v^2 = 2g(h - d) Substituting this expression for v into the equation for h, we get: h = (1/k) (1/2 M (2g(h - d))) = (1/k) (Mg(h - d)) Solving for h, we get: h = (kd/(2Mg)) + d Therefore, the answer is B, kd²^{/(2Mg) − d}.

A Blob of Clay of Mass M Is Propelled Upward