Numerical/Problem-solving
1. TITLE
How to solve River-Swimmer problems in NEET — Step-by-step
2. PROBLEM TYPE INTRO
This is a 2D Relative Velocity numerical. We use vector addition and vector resolution to find the resultant motion. A typical mistake students make is confusing the condition for the shortest path (zero drift) with the shortest time, or making sign errors when establishing components.
3. MCQ (Full numerical problem)
Question:
A swimmer can swim in still water with a speed of $5\text{ km/h}$. The river flows due East at a speed of $3\text{ km/h}$. At what angle (with respect to the perpendicular to the river flow) must the swimmer head to cross the river along the shortest path?
(A) $37^\circ$ upstream
(B) $53^\circ$ upstream
(C) $0^\circ$ (straight across)
(D) $37^\circ$ downstream
4. STEP-BY-STEP SOLUTION
SOLUTION
Given:
- Speed of swimmer in still water ($v_s$) = $5\text{ km/h}$
- Speed of river flow ($v_r$) = $3\text{ km/h}$
To Find:
- Angle ($\theta$) with the perpendicular to the river flow for the shortest path.
Formula:
- $v_x = v_s \sin \theta - v_r = 0$
Use this formula when you need zero horizontal drift (the shortest path directly to the opposite bank). Vector addition principles dictate that the horizontal component of the swimmer's velocity must perfectly cancel out the river's velocity.
Step 1: Set up the vector components
Let the perpendicular direction to the bank be the y-axis ($\hat{j}$) and the river flow be along the positive x-axis ($\hat{i}$). For the shortest path, the resultant velocity vector must lie entirely along the y-axis.
Step 2: Equate horizontal components to zero
The swimmer must head upstream at an angle $\theta$. The horizontal component of their velocity opposing the river is $v_s \sin \theta$. To perfectly cancel the river's flow:
$v_s \sin \theta = v_r$
$5 \sin \theta = 3$
Step 3: Solve for the angle
$\sin \theta = 3/5$
$\theta = 37^\circ$
Answer: $37^\circ$ upstream
5. WHERE STUDENTS GO WRONG
List 2-3 common mistakes:
- ❌ Mistake 1: Confusing "Shortest Path" with "Shortest Time." If the question asked for the shortest time, the swimmer should head straight across at $0^\circ$ (Option C).
- ❌ Mistake 2: Using the wrong trigonometric ratio. Students often write $\cos \theta = 3/5 \implies 53^\circ$ (Option B) when resolving components relative to the vertical axis.
- ❌ Mistake 3: Aiming downstream instead of upstream. Choosing Option D fails to recognize that relative velocity vector subtraction requires swimming against the current to cancel it out.
6. QUICK FORMULA REFERENCE
Formulas Used:
Formula 1 — Use $\sin \theta = \frac{v_{river}}{v_{swimmer}}$ when calculating the angle (from the perpendicular) required for the shortest path (zero drift).
Formula 2 — Use $\theta = 0^\circ$ (and $t = \frac{\text{width}}{v_{swimmer}}$) when finding the condition for the shortest time to cross.
7. SIMILAR PROBLEM HINT
"If NEET changes the values, the method stays the same. Practice with: a river flowing at $4\text{ km/h}$ and a swimmer speed of $8\text{ km/h}$. The component balancing condition $v_s \sin \theta = v_r$ works every time!"
8. NCERT REFERENCE
Based on: NCERT Class 11, Chapter 3, Example 3.3
9. CTA
More solved numericals → Kinematics Complete Guide
Practice problems → Mock exams