A small block slides down a frictionless track whose shape is described by y=x^2

Question

A small block slides down a frictionless track whose shape is described by y=x^2/d for x<0 and y=-x^2/d for x>0. The value of d is 0.3 meters, and x and y are measured in meters as usual. a) Suppose the block starts from rest on the track, at x=-3.1 meters. What will the block's speed be when it reaches x=0? b) Suppose the block starts on the track at x=0, and is given an initial velocity of 2.2m/s to the left. The block then begins to slide up the track to teh left. At what value of x will the block turn around and begin to slide down again? c) Now suppose the blocks starts on teh track at x=1.3m. The block is given a push to the left and begins to slide up the track, eventually reaching its maximum height at x=0, at which point it turns around and begins sliding down. What was its initial velocity in this case? d) Suppose the block starts on the track at x=0. What minimum initial velocity (moving to the right) must the block have such that it will leave the track at x=0 and go into freefall? e)You start the block on the track at rest, somewhere to the left of x=0. You then release the block from rest and let it slide down. What is the maximum value of x from which you can release the block from rest and have it leace the track at x=0 and go into freefall? (note: your answer should be a negative number, since you're starting to the left of the origin.)

Explanation / Answer

a) at x = -3.1 m, y = 32.033 m
   PE lost = KE gained
   g*y = 0.5v^2
   v = 25.05 m/s
b) Initial KE = 0.5*m*2.2^2 = 2.42 m = final PE = m*9.8*h
   h = 0.2469 m
   x = sqroot(d*h) = 0.27 m
c) initial velocity = v
   0.5*v^2 = 9.8*1.3^2/0.3
   v = 10.50 m/s
d) at x = 0
   dy/dx = 2x/d
   d^2y/dx^2 = 2/d
   r =( ()1+4x^2/d)^3/2) / (2/d)
   to leave the plank at any point
   mg*cos(theta) = mv^2/( ()1+4x^2/d)^3/2) / (2/d)
   put x = 0
   theta = 0
   g = 2v^2/d(1/d)^3/2
   v = 2.99 m/s

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.