Consider the two statements S₁ and S₂ given below and choose the correct option.

S₁ - Non-zero work has to be done on a moving particle to change its momentum.

S₂ - To change momentum of a particle, a non-zero net force should act on it.

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) or surrounded by
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. Let's construct the output precisely. Clean answer: `S<sub>1</sub> is false but S<sub>2</sub> is true.` Solution text structure:

The correct option is S₁ is false but S₂ is true.

Analysis of Statement S₂:
According to Newton's second law of motion, force is equal to the rate of change of momentum:

$\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$

For the momentum of a particle to change, the rate of change of momentum must be non-zero ($\frac{d\overrightarrow{p}}{dt}\ne 0$). This implies that a non-zero net force ($\overrightarrow{F}\ne 0$) must act on the particle.
Therefore, statement S₂ is true.

Analysis of Statement S₁:
Momentum ($\overrightarrow{p}=m\overrightarrow{v}$) is a vector quantity possessing both magnitude and direction. A change in momentum can occur by changing the speed (magnitude) or changing the direction of motion.

Work done on a particle is given by the work-energy theorem:

$W=\Delta K$

where $\Delta K$ is the change in kinetic energy of the particle.

Consider uniform circular motion: a particle moves along a circular path with constant speed. Since the direction of velocity constantly changes, its momentum changes continuously. However, the net force acting on it (centripetal force) is always perpendicular to its displacement ($\overrightarrow{F}\perp d\overrightarrow{r}$), so the work done on the particle is zero:

$W=\int \overrightarrow{F}·d\overrightarrow{r}=0$

Thus, it is possible to change the momentum of a particle with zero work done. Therefore, statement S₁ is false.

Hence, S₁ is false but S₂ is true.

Let's double-check all formatting constraints: - NO display="block" attribute on any $tag.\; -\; NO\; LaTeX.\; -\; NO\; raw\; html\; entities\; like\; -.\; -\; Every\; line\; containing$ block\; is\; placed\; in\; its\; own$$

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paragraph. - Both and tags are present. Everything is perfect. S₁ is false but S₂ is true.

The correct option is S₁ is false but S₂ is true.

Analysis of Statement S₂:
According to Newton's second law of motion, the net force acting on an object is equal to the time rate of change of its linear momentum:

$\overrightarrow{F}=\frac{d\overrightarrow{p}}{dt}$

To produce a change in momentum ($d\overrightarrow{p}\ne 0$), the rate of change of momentum must be non-zero. Consequently, a non-zero net external force ($\overrightarrow{F}\ne 0$) must act on the particle.
Therefore, S₂ is true.

Analysis of Statement S₁:
Linear momentum ($\overrightarrow{p}=m\overrightarrow{v}$) is a vector quantity. A change in momentum occurs when either the magnitude (speed) or the direction of the velocity vector changes.

According to the Work-Energy Theorem, the net work done on a particle is equal to the change in its kinetic energy:

$W=\Delta K=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

Consider a particle undergoing uniform circular motion. The direction of its velocity changes continuously, meaning its momentum is constantly changing. However, the net force (centripetal force) acting on the particle is always perpendicular to its displacement ($\overrightarrow{F}\perp d\overrightarrow{r}$). The work done by this force is:

$W=\int \overrightarrow{F}·d\overrightarrow{r}=0$

Since kinetic energy remains constant in uniform circular motion, the work done on the moving particle is zero even though its momentum changes. Thus, non-zero work is not required to change a particle's momentum.
Therefore, S₁ is false.

Conclusion:
S₁ is false but S₂ is true.