IITJEE: Projectile Motion & River-Boat Relative Motion Visualizer

Ashish Kumar Bahekar

Published on

October 7, 2026

CLASS 7

Apex Class • IITJEE 2D Kinematics Lab

Projectile Motion & River-Boat Relative Motion Visualizer

Switch between the Projectile Trajectory Lab and River-Boat Crossing Lab with one tap!

🎯 1. Projectile Motion Lab (u = 40 m/s) 🚤 2. River-Boat Relative Motion Lab

🚀 Select Launch Angle (θ) at Fixed Speed u = 40 m/s (g = 10 m/s²):

θ = 30° (Low Arc) θ = 45° (Max Range 🏆) θ = 60° (Same R as 30°!) θ = 75° (Steep Arc)

Time of Flight (T = 2u sinθ / g)

4.00 sec 5.66 sec (4√2 s) 6.93 sec (4√3 s) 7.73 sec

Depends ONLY on u_y = u sinθ

Max Height (H = u²sin²θ / 2g)

20.0 m 40.0 m 60.0 m 74.6 m

At Peak: v_y = 0, a = -g ↓

Horizontal Range (R = u²sin2θ / g)

138.6 m (80√3 m) 160.0 m (MAX 🏆) 138.6 m (Same as 30°!) 80.0 m (Same as 15°!)

Same for θ and (90° – θ)

Speed & KE at Highest Point

v = 34.6 m/s (¾ K₀) v = 28.3 m/s (½ K₀) v = 20.0 m/s (¼ K₀) v = 10.4 m/s

K_top = K₀ cos²θ (Never Zero!)

🔥 3 Golden JEE Projectile Shortcuts:
1. Relation between R & H: R tanθ = 4H (So at θ = 45°, Range is always 4 times Max Height: R = 4H).
2. Complementary Angles (θ & 90°-θ): Equal Range R₁ = R₂, Ratio of Heights H₁/H₂ = tan²θ, and R = ½ g T₁T₂.
3. Equation of Trajectory: y = x tanθ (1 – x/R) — use this directly whenever time t is not given!

🚤 River Width W = 120 m | Boat Speed v_br = 5 m/s | River Flow v_r = 3 m/s (→):

1. Shortest Time (Aim ⊥ 90°) 2. Shortest Path / Zero Drift (Aim 37° Upstream) 3. JEE Trap: What if River v_r > Boat v_br?

Crossing Velocity (v_y)

5.0 m/s (Full v_br) 4.0 m/s (√(5² – 3²)) v_br cosθ

Component ⊥ to River Bank

Crossing Time (t = W / v_y)

24.0 sec (FASTEST 🏆) 30.0 sec (120 / 4) Depends on Angle θ

Independent of River Speed in Case 1!

Downstream Drift (x)

72.0 m Downstream 0.0 m (ZERO DRIFT 🏆) Never Zero (v_r > v_br)

x = (v_r – v_br sinθ) × t

Steering Condition

θ = 0° (90° to Bank) sinθ = v_r / v_br = 3/5 sinθ = v_br / v_r (Min Drift)

Angle θ measured from Normal

💡 1-Second JEE Distinction (Shortest Time vs. Shortest Path):
• For Shortest Time: Always point the boat straight across (90° to bank) → t_min = W / v_br (even if the river is flowing at 100 m/s, crossing time does not change!).
• For Shortest Path (Zero Drift): Point upstream at angle θ from the normal where sinθ = v_r / v_br → t = W / √(v_br² – v_r²).