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Conservation of Energy

AP Physics 1· difficulty 4/5

θ μ_k

A spring of stiffness kk compressed by xx launches a block of mass mm along a horizontal frictionless surface and then up a rough incline of angle θ\theta with kinetic friction coefficient μk\mu_k. What is the maximum distance LL traveled along the incline before the block stops?

  • A

    L=kx22mgsin⁡θL=\dfrac{kx^2}{2mg\sin\theta}

  • B

    L=kx22μkmgcos⁡θL=\dfrac{kx^2}{2\mu_k mg\cos\theta}

  • C

    L=kxmg(sin⁡θ+μkcos⁡θ)L=\dfrac{kx}{mg(\sin\theta+\mu_k\cos\theta)}

  • D

    L=kx22mg(sin⁡θ+μkcos⁡θ)L=\dfrac{kx^2}{2mg(\sin\theta+\mu_k\cos\theta)}

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Explanation

Energy conservation with friction loss: 12kx2=mgLsin⁡θ+μkmgcos⁡θ L\tfrac{1}{2}kx^2 = mgL\sin\theta + \mu_k mg\cos\theta\,L. Solving gives L=kx22mg(sin⁡θ+μkcos⁡θ)L=\dfrac{kx^2}{2mg(\sin\theta+\mu_k\cos\theta)}.

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