Question

A particle slides down a frictionless parabolic $(y=x^2)$ track $(A-B-C)$ starting from rest at point $A$ (Figure). 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 is the below statement true or false?



(i) $KE$ at $P=KE$ at $B$

(ii) Height at $P$=height at $A$

(iii) Total energy at $P$ = total energy at $A$

(iv) Time of travel from $A\text{to}B$ = time of travel from (B~\text{to}~P\).

Answer 1

(i)

The particle slides on a frictionless parabolic track so the total energy of the particle will remain constant throughout the journey.

total energy at $A$ =total energy at $B$=total energy at $P$

At $B$ particle have only $KE$ but at $P$ some $KE$ is converted into $PE$

So, $(KE{)}_B>(KE{)}_P$

Final Answer: False


(ii)

As particle starts motion from point $A$. So, at $A$ particle has only potential energy.
This potential energy is converted into kinetic energy and potential energy at $P$.

Hence. $(PE{)}_P<(PE{)}_A$
Therefore, height at $P$<height at $A$

Final Answer : False


(iii)

As friction is absent in this track, total energy of the particle will remain constant throughout the journey.

Therefore, total energy at $P$ =total energy at $A$

Final Answer: true


(iv)

As we have, height at $P$< height at $A$,
and acceleration due to $g$ will be same in both the cases
Therefore, time of travel from $A\text{to}B\ne \text{time of travel from}B\text{to}P$
Final answer: False