Velocity


Let [x(t),y(t)] , t Î [a,b] be a parametric curve. The rate of change of the position [x(t),y(t)] is given by [x¢(t), y¢(t)] and is called the velocity at [x(t), y(t)]. The computer may help us understand the velocity by drawing the line segments joining [x(t),y(t)] with [x(t)+x¢(t), y(t)+y¢(t)] as t ranges over equally spaced points of the interval [a,b]. For example, the velocity along the circle [cos t, sin t], t Î [0,2p] is represented as:
The procedure follows these steps in Derive:
x: = cos(t) 
y: = sin(t) 
m: = [[x,y],[x+dif(x,t),y+dif(y,t)]] 
VECTOR(m,t,0,2pi,pi/50)
Then apply approX command to the last expression. Before plotting, make sure that Plot-Option-State-Mode is selected to Connected. In order to have all line segments drawn with the same color, be sure that Plot-Color-Auto is set to NO.

The velocity along the epicycloid [2 cos t-cos 2t, 2 sin t-sin 2t] appears as:

The velocity along the epicycloid [3 cos t-cos 3t, 3 sin t-sin 3t] appears as:
The velocity along the cycloid [t-sint,1-cos  t]  appears as: