Motion

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The location of a particle at time t is represented by the position vector x(t), which gives the location of the particle with respect to a fixed origin.

The velocity of the particle P is the vector v = dx(t) / dt = x'(t). This can be expressed in component form.

The speed of the particle P is v = |v| = |x'(t)|.

The acceleration of the particle P is given by a = dv/dt = x''(t).

Motion with Constant Velocity

Suppose that a particle P, initially at the point x0 with respect to a fixed origin, is moving with constant velocity v0, then the equation of motion of the particle is dx(t)/dt = v0. Integrating this gives the position of the particle at any time as x(t) = v0t + x0.

Motion with Constant Acceleration

The particle moving with constant acceleration a, giving x''(t) = a. Integrating once gives dx/dt = v(t) = at + u, where u is the initial velocity of the particle.

Integrating again gives the position vector of the particle as x(t) = (at^2)/2 + ut + d, where d is a constant vector denoting the inital position of the particle.

Motion in a Straight Line

In the case of motion in a straight line, the position of a particle P is x = x(t)n where n is a vector of unit length in the direction of the straight line. Similarly, the velocity and acceleration of P are respectively v = x'(t)n and a = x''(t)n.

The equivalent expressions for s(t), the displacement from the starting position, and v(t) for a particle moving with acceleration a = an, are

s = ut + (a/2) t^2, v = u + at, v^2 = u^2 + 2as