Homework SourseStarting at the origin a particle moves along a parabolic cu
Starting at the origin, a particle moves along a parabolic curve y=x^2 in the first quadrant. An observer positioned at A(0,1) watches as the particle moves from the origin to the point P(3,9). During this time, what is the shortest distance between the particle and the observer?
Solution
Let us assume a parametric equation for a point on the curve.
Let x = t and y = t2
Distance between , (0,1) and (t , t2) has to be minimised.
Thus,
s2 = k = (t - 0)2 + (t2 -1)2 has to be minimised.
putting d/dt = 0,
2t + 2(t2 -1) ( 2t) = 0
4t3 - 2t = 0
t ( 2t2 - 1) = 0
Thus, t = 0 , t = 1/ sqrt (2) since ( 0 < t < 3)
Thus, minimum distance = >
at t = 0, s = 1
at t = 1/sqrt (2) ,
s = sqrt ( 1/2 + 1/4 )
= 0.866
Thus, smallest distance = 0.866 = sqrt (3) / 2
Hope this helps.
