Classical mechanics problem for a free particle

In summary, the problem involves a free particle with mass m moving in one space dimension with an initial velocity v0 and a starting point of x = 0 at time t = 0. The particle's end point is at x = v0t1 at time t = t1. The task is to calculate the action I for this path. In part b, the particle has the same initial and final points, but with a different initial velocity v1 and a nonzero constant acceleration a. The goal is to find a as a function of v1 and determine the trajectory x(t), which will not satisfy the equations of motion since there is no potential energy. In part c, the action I(v1) is to be found for
  • #1
becks1
1
0
Summary: The initial problem states: Consider a free particle of mass m moving in one space dimension with velocity v0. Its
starting point is at x = x0 = 0 at time t = t0 = 0 and its end point is at x = x1 = v0t1
at time t = t1 > 0. and this info is to do the 3 problems written out.

a) Calculate the action I for this path.
b) Now suppose that the particle has the same initial and final points in space and
time, but now has an initial velocity v1 6= v0 and a nonzero constant acceleration
a. Find a as a function of v1 and give the trajectory x(t). Note that this is still
a free particle with no potential energy, so x(t) will not satisfy the equations of
motion.
c) Find the action I(v1) for this trajectory, Show that dI/dv1 = 0 for the path
which solves the equation of motion. Is this path a maximum, a minimum. or an
inflection point of I(v1)?
 
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  • #3
@becks1, please give relevant equations and show some attempt at a solution.
 
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FAQ: Classical mechanics problem for a free particle

What is a free particle in classical mechanics?

A free particle in classical mechanics is an object that is not influenced by any external forces. This means that it is not subject to any forces such as gravity or friction, and its motion is determined solely by its initial position and velocity.

What is the equation of motion for a free particle?

The equation of motion for a free particle in classical mechanics is given by Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In mathematical terms, this can be expressed as F = ma, where F is the net force, m is the mass of the particle, and a is its acceleration.

How is the motion of a free particle described in classical mechanics?

In classical mechanics, the motion of a free particle is described using the laws of motion developed by Isaac Newton. These laws state that an object will remain at rest or continue moving at a constant velocity unless acted upon by an external force, and that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

What is the principle of conservation of energy in classical mechanics?

The principle of conservation of energy in classical mechanics states that the total energy of a closed system remains constant over time. This means that energy can neither be created nor destroyed, but can only be transferred from one form to another. In the case of a free particle, its total energy is equal to its kinetic energy, which is determined by its mass and velocity.

How is the trajectory of a free particle determined in classical mechanics?

In classical mechanics, the trajectory of a free particle can be determined using the equations of motion and the initial conditions of the particle. By solving for the position and velocity of the particle at different points in time, the path or trajectory that the particle follows can be determined. This trajectory will be a straight line if the particle is moving at a constant velocity, or a curved line if the particle is accelerating.

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