Question

A particle slides down a frictionless parabolic
($y=x^2$) track (A – B – C) starting from rest at
point A (Fig.). Point B is at the vertex of
parabola and point C is at a height less than
that of point A. After C, the particle moves freely
in air as a projectile. If the particle reaches
highest point at P, then

• A. KE at P = KE at B
• B. height at P = height at A
• C. total energy at P = total energy at A
• D. time of travel from A to B = time of travel from
  B to P

Answer 1

The correct option is C total energy at P = total energy at A

In such type of problems, we have to observe the nature of track that if there is a friction or not, as friction is not present in this track, total energy of the particle will remain constant throughout the journey.

According to the problem, the path traversed by the particle on a friction less track is parabolic, is given by the equation y = x2, thus total energy (KE + PE) will be same throughout the journey.

Hence, total energy at A = total energy at P

At B the particle is having only KE but at P some KE is converted to PE.

So, (KE)B > (KE)P

Total energy at A = PE = Total energy at B = KE = Total energy at P = PE + KE

The potential energy at A is converted to KE and PE at P, hence (PE)P < (PE)A

Hence, (Height) P < (Height) A

As, Height of P < Height of A

Hence, path length AB > path length BP

Hence, time of travel from A to B ≠ Time of travel from B to P.