Two particles are projected vertically and horizontally respectively from end points A and B of an inclined plane as shown in the figure. Length of the incline is ℓ and particle A is projected vertically upwards with velocity v₀. Both the particles are projected at the same time and they collide in mid-air. Velocity with which B is projected is v₀√n. Then find n.
Physics · Mixed Chapter
JEE Mains

Question

Two particles are projected vertically and horizontally respectively from end points A and B of an inclined plane as shown in the figure. Length of the incline is ℓ and particle A is projected vertically upwards with velocity v₀. Both the particles are projected at the same time and they collide in mid-air. Velocity with which B is projected is v₀√n. Then find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: Set A at (0,0), B at (ℓcos30°, ℓsin30°). Both projectiles experience the same downward gravitational acceleration, so their relative acceleration is zero. Step 2: Relative vertical motion requires v₀t=ℓsin30°=ℓ/2, giving t=ℓ/(2v₀). Step 3: If B is launched horizontally leftwards at speed u, horizontal meeting requires ut=ℓcos30°=√3ℓ/2. Step 4: u=(√3ℓ/2)/(ℓ/(2v₀))=√3v₀=v₀√n. Hence n=3.