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Newton's laws are often stated in terms of point or particle masses, that is, bodies whose volume is negligible. This is a reasonable approximation for real bodies when the motion of internal parts can be neglected, and when the separation between bodies is much larger than the size of each.
The relativistic energy–momentum relationship holds even for massless particles such as photons; by setting m 0 = 0 it follows that =. In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic collision, the paths formed by the two afterwards will form an acute angle.
There are two main descriptions of motion: dynamics and kinematics.Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
Proper acceleration at any speed is the physical acceleration experienced locally by an object. In spacetime it is a three-vector acceleration with respect to the object's instantaneously varying free-float frame. [13] Its magnitude α is the frame-invariant magnitude of that object's four-acceleration. Proper acceleration is also useful from ...
Proper acceleration reduces to coordinate acceleration in an inertial coordinate system in flat spacetime (i.e. in the absence of gravity), provided the magnitude of the object's proper-velocity [3] (momentum per unit mass) is much less than the speed of light c.
The corresponding three-acceleration = (, , ) in these frames can be directly measured by an accelerometer, and is called proper acceleration [18] [H 14] or rest acceleration. [19] [H 12] The relation of in a momentary inertial frame ′ and measured in an external inertial frame follows from (1c, 1d) with ′ =, ′ =, = and =.
By the fundamental theorem of calculus, it can be seen that the integral of the acceleration function a(t) is the velocity function v(t); that is, the area under the curve of an acceleration vs. time (a vs. t) graph corresponds to the change of velocity. =.
A rocket's required mass ratio as a function of effective exhaust velocity ratio. The classical rocket equation, or ideal rocket equation is a mathematical equation that describes the motion of vehicles that follow the basic principle of a rocket: a device that can apply acceleration to itself using thrust by expelling part of its mass with high velocity and can thereby move due to the ...