Motion of Particles
Mechanics is the study of the physical laws that describe the motion of objects. When studying moving objects, it is often useful to simplify their motion by treating them as single particles. In this context, a particle is defined as a small object whose microscopic structure is irrelevant to its motion. The fundamental concepts of mechanics are thus all introduced using this simplification, from kinematics, the branch of mechanics that quantitatively describes how a particle moves, to dynamics, which studies motion and its causes.
The simplest case of motion is treated by considering the rectilinear or translational motion of a particle along an axis. This type of motion is described using the following quantities: displacement, speed, velocity and acceleration. The position of a particle in space is a vector quantity because it has both magnitude and direction since it is specified according to a reference point. When a particle moves, its position changes as a function of time by a given distance, d. A change of position is also a vector quantity and it is called displacement, d. If the particle moves from x at time t to x at time t, then its average velocity, or the rate at which it changes its position in the time interval t = t-t, is equal to its displacement divided by the time interval or v = d</t and its average speed is the distance traveled in the same time interval. The instantaneous velocity is the velocity of the particle at an instant in time, v = dr/dt where dr is the displacement during the time interval t. The instantaneous speed is the magnitude component of the instantaneous velocity vector. The acceleration of a particle is defined as the rate at which the particle changes its instantaneous velocity. If at time t the particle has a velocity v, and at time t it has a velocity v, then the change in instantaneous velocity is v = v-v in the time interval t = t-t and the average acceleration (a) during that time interval is a = v/t. The instantaneous acceleration is a = dv/dt or the change in instantaneous velocity in the time interval t.
The motion of a particle in one dimension is easily represented using diagrams which plot the relevant quantities. For example, if a particle is moving with a constant, rightward (positive) velocity, the position vs. time diagram will show a straight line of constant and positive slope. If the particle is moving with a rightward (positive) and changing velocity, the position vs. time diagram will result in an upward curving line. This allows one to distinguish between constant velocity and accelerated motion. The rectilinear motion of a particle with constant acceleration is also described using kinematic equations showing how its velocity and displacement change. Thus, for one-dimensional constant acceleration, the velocity varies with time according to v = v + at, the displacement varies with time as x = x + vt + ½at[sup2 ] and the velocity varies with position according to v[sup2 ] = v[sup2 ] + 2a(x - x).
The three-dimensional motion of a particle with constant acceleration is described by including all three components, x, y and z, of the vector quantities. If acceleration is constant, then: a = aî + aĵ + ak. The components a, a, and a are also constant and the average acceleration equals the instantaneous acceleration. If at t = 0, the particle is at position r = xî + yĵ + zk and has velocity v = vî + vĵ + vk, then, at time t, its velocity has changed by an amount v = at = at or, in terms of the vector equation incorporating the three components: vî + vĵ + vk = atî + atĵ + ak, and yields the following set describing each component for each dimension: v = at, v = at, and v = at. Then, at time t, v = v + v = v + at, v = v + at, v = v + at, or v = v + at.
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