JEE PYQ: Motion in a Plane - Question ID cc57fb625565 (JEE Main 2025)

Core Formula & Concept:

In this problem we combine two key ideas from “Motion in a Plane”:

1. Relative velocity in two dimensions When a particle is projected from a moving platform, its initial velocity relative to the ground is the vector sum of the platform’s velocity and the projection velocity relative to the platform. If the boat moves downstream with speed $v_b$ and the river flows at $v_r$, then the boat’s speed relative to the bank is $v_{\text{boat, bank}} = v_b + v_r.$ 2. Projectile motion A ball thrown vertically upward with speed $u$ relative to the boat will, relative to the bank, have an initial horizontal component $v_x = v_{\text{boat, bank}}$ and an initial vertical component $u_y = u$. The time of flight $T$ for a projectile launched and landing at the same vertical level is $T = \frac{2\,u_y}{g}.$ The horizontal range $R$ is then $R = v_x \cdot T.$ Step-by-Step Derivation:

Step 1: Convert all speeds to SI units (m/s)

• Boat’s maximum speed in still water: $27\;\text{km/h} = 27\times\frac{1000}{3600} = 7.5\;\text{m/s}.$
• River flow speed: $9\;\text{km/h} = 9\times\frac{1000}{3600} = 2.5\;\text{m/s}.$
• Ball’s projection speed (already in m/s): $u = 10\;\text{m/s}.$

Step 2: Determine the boat’s speed relative to the bank

Since the boat is moving downstream, its speed relative to the bank is the sum of its speed in still water and the river’s speed: $v_{\text{boat, bank}} = 7.5 + 2.5 = 10\;\text{m/s}.$

Step 3: Resolve the ball’s initial velocity relative to the bank

• Horizontal component: $v_x = v_{\text{boat, bank}} = 10\;\text{m/s}.$
• Vertical component: $u_y = 10\;\text{m/s}.$

Step 4: Compute the time of flight

The ball goes up and returns to the same vertical level, so $T = \frac{2\,u_y}{g} = \frac{2\times10}{10} = 2\;\text{s}.$

Step 5: Calculate the horizontal range

The range is the horizontal distance covered in time $T$: $R = v_x \cdot T = 10\;\text{m/s}\times 2\;\text{s} = 20\;\text{m}.$ Convert to centimetres: $20\;\text{m} = 20\times100 = 2000\;\text{cm}.$ Common Traps & Exam Tip:

1. Unit inconsistency: Students often forget to convert km/h to m/s, leading to incorrect numerical results. 2. Relative velocity confusion: One must add the boat’s speed in still water and the river’s speed to get the boat’s speed relative to the bank. 3. Vertical vs horizontal independence: The vertical motion determines the time of flight; the horizontal motion then uses that time to find the range. 4. Sign of projection: Throwing “vertically upwards” means the vertical component is positive; no need to introduce negative signs prematurely.

Exam tip: Always draw a quick sketch of the velocity vectors and label the reference frames (boat vs bank). This clarifies which velocities add and which remain separate.