In FIGURE EX10.28, what is the maximum speed a 200 g particle could have at x = 2.0 m and never reach x = 6.0 m?

Question

Graph showing potential energy (U) vs position (x) for a particle in a physics problem.

Accepted Answer

Hey everyone. So this problem is dealing with potential energy graphs. Let's see what it's asking us if a particle of mass 450 g is positioned at X equals five m and must never surpass the point at X equals eight m. What is the highest speed it can achieve along its trajectory? We are told to use the law of conservation energy to determine the maximum speed that a particle should have. So our multiple choice answers here are a 3.42 m per second. B 8.21 m per second. C 7.78 m per second or D 5.96 m per second. So when we take a look at our potential energy graph, We can see that there is this apex, this mountain that happens at seven m. And so for us to never reach, never surpass the point at X equals eight m, we must have zero kinetic energy at this seven at this peak at seven m. And so we can recall that our total energy E is equal to our potential energy plus our kinetic energy. This peak at seven m has a potential energy of eight jewels. So we know that B equals eight jewels plus zero jewels Because we've established that the kind of energy must be zero at seven m. We can also recall that the kinetic energy Is given by an equation of 1/2 M V squared. And so the hint to use the law of conservation of energy is a hint that allows us to recall that our highest speed or V max happens when our potential energy is zero. And so in that scenario, so when we get to V Max or in the case of V Max, we would have our total energy, right, the law of conservation of energy is going to be the same. And in this case, we are going to have our potential energy equal zero. And so Our E which we've already determined is eight Jews. We'll have eight jewels. OK. That's a little messy there. We'll have eight jewels equals K. It's a K is our kinetic energy when we have V Max. So we can plug that into this equation here and we'll have eight Jes equals one half multiplied by the mass, Which from our problem was given as 450 g. We can rewrite that as 0.4, five kg to keep it in standard units time multiplied by V max squared. And so when we rearrange that equation to solve for V Max, we get V Max is equal to the square root of two multiplied by eight jewels All divided by 0.45 kg. And we plug that into our calculator and we get 5.9, m/s. So that is our answer for this problem. And that aligns with answer choice D, so that's all we have for this one. We'll see you in the next video.

In FIGURE EX10.28, what is the maximum speed a 200 g particle could have at x = 2.0 m and never reach x = 6.0 m?

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Key Concepts

Potential Energy (U)

Conservation of Energy

Kinetic Energy (KE)