EGR 280 Mechanics 8 – Particle Kinematics I

Dynamics Two distinct parts: Kinematics – concerned with the mathematics that describe the geometry of motion, without being concerned with why that motion takes place. Kinetics – the study of the relationships between the forces that act on a body and its resulting motion.

Rectilinear motion of particles (motion along a straight line) Keep track of the position of the particle with a position coordinate x: The time rate of change of the position is velocity: v = dx/dt The time rate of change of the velocity is acceleration: a = dv/dt and a = dv/dt = d(dx/dt)/dt = d 2 x/dt 2 ; a = d 2 x/dt 2 a = dv/dt = (dv/dx)(dx/dt) = v(dv/dx); a = v(dv/dx) O x

3 classes of problems: 1.Acceleration a is given as a function of time: integrate: v = ∫a dt; x = ∫v dt 2.Acceleration is given as a function of position: integrate: ∫v dv = ∫a dx 3.Acceleration is given as a function of velocity: integrate: ∫dt = ∫(1/a)dv or ∫dx = ∫(v/a)dv

If we know the acceleration not as an analytical function, but as discrete values in time, we can still integrate these discrete values to find the velocity, and subsequently integrate the velocity to find the position. Consider the function below: The integral of f(x) can be piecewise-linearly approximated as: This is known as the Trapezoidal Rule, and has local error of order (Δt) 2, so the smaller Δt is, the more accurately the numerical integration be performed. f(t) ΔtΔt fifi f i+1 t

Uniform rectilinear motion – when acceleration is zero a = 0 = dv/dtvelocity is constant v 0 = dx/dt x = x 0 + v 0 t Uniformly accelerated rectilinear motion –when acceleration is constant a 0 = dv/dt v = v 0 + a 0 t v = dx/dt x = x 0 + v 0 t + ½ a 0 t 2 a 0 = v(dv/dx) v 2 = v a 0 (x-x 0 )

Motion of several particles x AB is the position from A to B x B = x A + x AB x B/A is the position of B with respect to A x B = x A + x B/A Taking time derivatives: v B = v A + v B/A v B = v A + v AB a B = a A + a B/A a B = a A + a AB 0 xAxA A B x AB = x B/A xBxB