Question

Column I describes some situations in which a small object moves. Column II describes some characteristics of these motions. Match the situations in Column I with the characteristics in Column II:

Column I		Column II
A	The object moves on the $x$-axis under a conservative force in such a way that its "speed" and "position" satisfy $v=c_1\sqrt{c_2-x^2}$, where $c_1$, and $c_2$ are positive constants.	P	The object executes a simple harmonic motion.
B	The object moves on the $x$-axis in such a way that its velocity and its displacement from the origin satisfy $v=-kx$, where $k$ is a positive constant.	Q	The object does not change its direction.
C	The object is attached to one end of a mass-less spring of a given spring constant. The other end of the spring is attached to the ceiling of an elevator. Initially everything is at rest. The elevator starts going upwards with a constant acceleration a. The motion of the object is observed from the elevator during the period it maintains this acceleration.	R	The kinetic energy of the object keeps on decreasing.
D	The object is projected from the earth's surface vertically upwards with a speed $2\sqrt{G{M}_e/R_e}$, where $M_e$ is the mass of the Earth and $R_e$ is the radius of the Earth. Neglect forces from objects other than the Earth.	S	The object can change its direction only once.

• A. $(A)\to (P);(B)\to (Q),(S);(C)\to (P);(D)\to (Q),(R)$
• B. $(A)\to (P);(B)\to (Q),(R);(C)\to (P);(D)\to (Q),(S)$
• C. $(A)\to (P);(B)\to (Q),(R);(C)\to (P);(D)\to (Q),(R)$
• D. $(A)\to (P);(B)\to (P),(R);(C)\to (P);(D)\to (Q),(R)$

Answer 1

Answer: (C)

$(A)\to (P)$
In SHM, $v=\omega \sqrt{a^2-x^2}$ which resembles
$v=c_1\sqrt{c_2-x^2}$
and hence the object executes simple harmonic motion.
(B) $\to (Q),(R)$
We have
$v=-kx=\frac{dx}{dt}\Rightarrow \int \frac{dx}{x}=-k\int dt$
That is,
In $(x)=-kt\Rightarrow x=e^{-kt}$
Therefore,
$v=\frac{dx}{dt}=-k{e}^{-kt}$
Now,
K.E. $=\frac{1}{2}m{v}^2=\frac{1}{2}m{k}^2{e}^{-2kt}$
That is, the object does not change its direction as shown in the following graph:

The kinetic energy keeps on decreasing as shown in the following graph:

$(C)\to (P)$
We have $T=2\pi \sqrt{\frac{m}{k}};$
therefore, the motion of the object is SHM.
(D) $\to (Q),(R)$
As the object goes up against gravity, its speed and hence the kinetic energy goes on decreasing. Also, since
$V=2\sqrt{\frac{G{M}_e}{R_e}}>$ Escape speed $=\sqrt{\frac{2G{M}_e}{R_e}}$
we conclude that the object cannot return to Earth once again.