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[2] The kinetic energy of an object is equal to the work, or force in the direction of motion times its displacement , needed to accelerate the object from rest to its given speed. The same amount of work is done by the object when decelerating from its current speed to a state of rest. [2]
Kinetic energy T is the energy of the system's motion and is a function only of the velocities v k, not the positions r k, nor time t, so T = T(v 1, v 2, ...). V , the potential energy of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any ...
The general equation can then be written as [6] = + + (),. where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions.
The kinetic energy of the system is: = (˙ + ˙) where is the mass of the bobs, is the length of the strings, and , are the angular displacements of the two bobs from equilibrium. The potential energy of the system is: E p = m g L ( 2 − cos θ 1 − cos θ 2 ) + 1 2 k L 2 ( θ 2 − θ 1 ) 2 {\displaystyle E_{\text{p}}=mgL(2-\cos ...
By contrast, subtracting equation (2) from equation (1) results in an equation that describes how the vector r = x 1 − x 2 between the masses changes with time. The solutions of these independent one-body problems can be combined to obtain the solutions for the trajectories x 1 (t) and x 2 (t).
The energy function in the action principles is not the total energy (conserved in an isolated system), but the Lagrangian, the difference between kinetic and potential energy. The kinetic energy combines the energy of motion for all the objects in the system; the potential energy depends upon the instantaneous position of the objects and ...
Hamilton's principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q 1, q 2, ..., q N) between two specified states q 1 = q(t 1) and q 2 = q(t 2) at two specified times t 1 and t 2 is a stationary point (a point where the variation is zero) of the action functional [] = ((), ˙ (),) where (, ˙,) is the Lagrangian function for the system.
In such a collision, kinetic energy is lost by bonding the two bodies together. This bonding energy usually results in a maximum kinetic energy loss of the system. It is necessary to consider conservation of momentum: (Note: In the sliding block example above, momentum of the two body system is only conserved if the surface has zero friction.