In this problem, we're going to use the equation for the Kinetic energy:

$\overline{){\mathbf{K}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{2}}{{\mathbf{mv}}}^{{\mathbf{2}}}}$

From the law of conservation of energy:

Initial K = Work done against friction + spring energy.

$\begin{array}{rcl}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}{\mathbf{v}}^{\mathbf{2}}& \mathbf{=}& \mathbf{\mu}\mathbf{m}\mathbf{g}\mathbf{x}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{k}{\mathbf{x}}^{\mathbf{2}}\end{array}$........................ (1)

An object of mass, m, is traveling on a horizontal surface. There is a coefficient of kinetic friction, μ, between the object and the surface. The object has speed, v, when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x=0 on its return trip, it stops. Find k, the spring constant. Express k in terms of m_{u} , m, g, and v.

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What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Work By Springs concept. You can view video lessons to learn Work By Springs. Or if you need more Work By Springs practice, you can also practice Work By Springs practice problems.