Learning Goal: To understand how to apply thelaw of conservation of energy to si

Question

Learning Goal: To understand how to apply thelaw of conservation of energy to situations with and withoutnonconservative forces acting.

The law of conservation of energy states the following:

where the quantities with subscript "i" refer to the "initial"moment and those with subscript "f" refer to the final moment. Awise choice of initial and final moments, which is not alwaysobvious, may significantly simplify the solution.

Potential energy, in contrast, has many forms. Two forms thatyou will be dealing with often are gravitational andelastic potential energy. Gravitational potential energyis the energy possessed by elevated objects. For small heights, itcan be found as

The gravitational force and the elastic force are two examplesof conservative forces. What if nonconservative forces, such asfriction, also act within the system? In that case, the totalmechanical energy will change. The law of conservation of energy isthen written as

You will analyze the motion of the block at different momentsusing the law of conservation of energy.

Explanation / Answer

If all you have to do is find the speed at the bottom of the ramp,you just need to sum up the different forms of energy that youhave. At the beginning, your speed is v. So the kinetic energy is(1/2)mv2 The gravitational potential energy is given by mgh, so your totalenergy is now (1/2)mv2 + mgh (h is the vertical height of the ramp) Now, all this energy is now in the form of kinetic energy, so wehave this: [Total energy]   =   [Final kinetic energy] (1/2)mv2 + mgh = (1/2)mvb2 Now solve for vb (you can factor out the m's) v2 + 2gh = vb2 vb = (v2 + 2gh) Edit: the post above me is close, but not quite right becausethe initial velocity squared was supposed to be in the squareroot.

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