JEE PYQ: Motion in a Plane - Question ID 8bb797b2d294 (JEE Main 2019)

To solve this problem, we analyze the motion of the swimmer in two dimensions—along the river (downstream) and perpendicular to it (across the river). The key concept is relative velocity in a plane. The swimmer’s velocity relative to the water ($\vec{v}_{sw}$) and the river’s velocity ($\vec{v}_{r}$) combine vectorially to give the swimmer’s resultant velocity relative to the ground ($\vec{v}_{sg}$).

The condition to "cross the river straight" means the swimmer’s resultant velocity must be purely perpendicular to the river banks—i.e., no component along the river. Mathematically, this requires:

For the resultant to have zero component along the river, the swimmer must swim at an angle $\theta$ such that the component of $\vec{v}_{sw}$ along the river exactly cancels $\vec{v}_{r}$.

1. Define Directions: Let the river flow along the positive x-axis. The swimmer aims to cross straight along the positive y-axis. Thus, the river velocity is: $\vec{v}_{r} = 2 \text{ km/h} \ \hat{i}$ The swimmer’s velocity relative to water is: $\vec{v}_{sw} = 4 \text{ km/h} \ (\cos\theta \ \hat{i} + \sin\theta \ \hat{j})$ where $\theta$ is the angle the swimmer makes with the river flow (x-axis).

2. Resultant Velocity Condition: The swimmer’s resultant velocity relative to ground is: $\vec{v}_{sg} = \vec{v}_{sw} + \vec{v}_{r} = (4\cos\theta + 2)\hat{i} + 4\sin\theta \ \hat{j}$ For crossing straight, the x-component must vanish: $4\cos\theta + 2 = 0$

3. Solve for $\theta$: $4\cos\theta = -2$ $\cos\theta = -\frac{1}{2}$ The angle whose cosine is $-\frac{1}{2}$ in the range $0° \leq \theta \leq 180°$ is: $\theta = 120°$

4. Interpretation: The swimmer must swim at $120°$ with respect to the river flow (x-axis) to cancel the downstream drift and cross straight.

- Misinterpreting the angle: Students often confuse whether the angle is measured with respect to the river flow or the perpendicular. Always define the reference direction clearly. - Sign errors in components: Forgetting that the swimmer’s x-component must oppose the river flow leads to incorrect angles. Double-check the vector addition. - Unit consistency: Ensure all speeds are in the same units (km/h here) to avoid calculation errors.

Exam Tip: Draw a clear velocity vector diagram. The swimmer’s velocity vector should form an obtuse angle with the river flow to counteract the downstream current.