Q57 / 270 · NDA 2026 (Phase 1) · General Ability
The position vector of a particle is given by $\vec{r} = \sqrt{3}\,t^2\hat{i} + \sqrt{2}\,t\hat{j} + \sqrt{5}\,\hat{k}$. Which one of the following statements is correct?
- AThe force acting on the particle is parallel to the instantaneous momentum of the particle.
- BThe force acting on the particle is perpendicular to the instantaneous momentum of the particle.
- CThe particle experiences zero force.
- Correct answerDThe force acting on the particle is not parallel to the instantaneous momentum of the particle.
Solution
Given position vector $\vec{r} = \sqrt{3}t^2\hat{i} + \sqrt{2}t\hat{j} + \sqrt{5}\hat{k}$
Velocity $\vec{v} = \dfrac{d\vec{r}}{dt} = 2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j}$
Instantaneous momentum $\vec{p} = m\vec{v} = m(2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j})$
Acceleration $\vec{a} = \dfrac{d\vec{v}}{dt} = 2\sqrt{3}\hat{i}$
Force $\vec{F} = m\vec{a} = 2\sqrt{3}m\hat{i}$
Taking the cross product of force and momentum:
$\vec{F} \times \vec{p} = (2\sqrt{3}m\hat{i}) \times m(2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j}) = 2\sqrt{6}m^2\hat{k} \neq 0$
Since the cross product is non-zero, the force is not parallel to the instantaneous momentum.
Also, $\vec{F} \cdot \vec{p} = 12m^2t \neq 0$, so they are not perpendicular either.
Answer: The force acting on the particle is not parallel to the instantaneous momentum of the particle.
Velocity $\vec{v} = \dfrac{d\vec{r}}{dt} = 2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j}$
Instantaneous momentum $\vec{p} = m\vec{v} = m(2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j})$
Acceleration $\vec{a} = \dfrac{d\vec{v}}{dt} = 2\sqrt{3}\hat{i}$
Force $\vec{F} = m\vec{a} = 2\sqrt{3}m\hat{i}$
Taking the cross product of force and momentum:
$\vec{F} \times \vec{p} = (2\sqrt{3}m\hat{i}) \times m(2\sqrt{3}t\hat{i} + \sqrt{2}\hat{j}) = 2\sqrt{6}m^2\hat{k} \neq 0$
Since the cross product is non-zero, the force is not parallel to the instantaneous momentum.
Also, $\vec{F} \cdot \vec{p} = 12m^2t \neq 0$, so they are not perpendicular either.
Answer: The force acting on the particle is not parallel to the instantaneous momentum of the particle.
Hence, the correct answer is option: 'D'.