2) A particle of mass 2/3 Kg is subjected to a potential energy function V(x) = (3x ^ 2 - 2x ^ 3) * J where x >= 0 and expressed in meters. Find the positions of all the maxima and minima. What is the maximum value of the potential energy?b) Supposing the particle is released at x = 4/3 * m find its velocity when it reaches x = 3/2 * n .?

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2) A particle of mass 2/3 Kg is subjected to a potential energy function V(x) = (3x ^ 2 - 2x ^ 3) * J where x >= 0 and expressed in meters. Find the positions of all the maxima and minima. What is the maximum value of the potential energy?
b) Supposing the particle is released at x = 4/3 * m find its velocity when it reaches x = 3/2 * n .?

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2) A particle of mass 2/3 Kg is subjected to a potential energy functi...

Maxima and Minima Positions:
- To find the maxima and minima positions, we need to calculate the first derivative of the potential energy function V(x).
- The first derivative of V(x) is given by V'(x) = dV(x)/dx = 6x - 6x^2.
- Setting V'(x) equal to zero and solving for x, we get x = 0 and x = 1 as critical points.
- To determine if these critical points are maxima or minima, we can use the second derivative test.
- The second derivative of V(x) is given by V''(x) = d^2V(x)/dx^2 = 6 - 12x.
- Evaluating V''(0) and V''(1), we find that V''(0) = 6 > 0 and V''(1) = -6 < />
- Therefore, x = 0 corresponds to a local minimum and x = 1 corresponds to a local maximum.

Maximum Value of Potential Energy:
- To find the maximum value of the potential energy, we substitute x = 1 into the potential energy function.
- V(1) = (3*1^2 - 2*1^3) = 3 - 2 = 1 J.
- Therefore, the maximum value of the potential energy is 1 Joule.

Velocity at x = 3/2:
- To determine the velocity of the particle at x = 3/2, we need to use the conservation of mechanical energy.
- The initial mechanical energy of the particle at x = 4/3 is given by Ei = V(4/3) + Ki, where Ki is the initial kinetic energy (which we assume to be zero).
- The final mechanical energy of the particle at x = 3/2 is given by Ef = V(3/2) + Kf, where Kf is the final kinetic energy.
- Since mechanical energy is conserved, Ei = Ef. Therefore, V(4/3) = V(3/2).
- By substituting x = 4/3 and x = 3/2 into the potential energy function, we can solve for the final kinetic energy Kf.
- Once Kf is determined, we can calculate the final velocity using the equation Kf = (1/2)mv^2, where m is the mass of the particle.

Answered by Notes Wala · View profile

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Passage 2Newtons surprising success at developing the laws of motion, as well as the development and refinement of other physical laws, led to the idea of scientific determinism. The first expression of this principle was in the beginning of the nineteenth century by Laplace, a French scientist. Laplace argued that if one knew the position and velocity of all the particles in the universe at a given time, the laws of physics would be able to predict the future state ofthe universe.Scientific determinism held sway over a great many scientists until the early twentieth century, when the quantum mechanics revolution occurred. Quantum mechanics introduced the world to the idea of the uncertainty principle, which stated that it was impossible to accurately measure both the position and the velocity of a particle at one time. Because Laplaces omniscience could never occur, even in theory, the principle of scientific determinism was thrown into doubt. However, quantum mechanics does allow for a reduced form of scientific determinism. Even though physicists are unable to know precisely where a particle is and what its velocity is, they can determine certain probabilities about its position and velocity. These probabilities are called wave functions. By use of a formula known as the Schrodinger equation, a scientist with the wave function of a particle at a given time can calculate the particles future wave function. These calculations can give the particles position or velocity, but not both. Thus, the physicist is in possession of exactly half ofthe information needed to satisfy Laplaces view ofdeterminism. Unfortunately, under modern physics theories, that is far as any researcher can go in predicting the future.Q. Which of the following, if true, would most strengthen the authors conclusion in the passages final sentence?

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2) A particle of mass 2/3 Kg is subjected to a potential energy function V(x) = (3x ^ 2 - 2x ^ 3) * J where x >= 0 and expressed in meters. Find the positions of all the maxima and minima. What is the maximum value of the potential energy?b) Supposing the particle is released at x = 4/3 * m find its velocity when it reaches x = 3/2 * n .?

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