A projectile is to be launched so as to pass over a hemispherical mountain. This can be accomplished if the projectile is launched at a certain minimum distance D from the centre of the mountain. What is minimum required D for r = √2 m?
Physics · Mixed Chapter
JEE Mains

Question

A projectile is to be launched so as to pass over a hemispherical mountain. This can be accomplished if the projectile is launched at a certain minimum distance D from the centre of the mountain. What is minimum required D for r = √2 m?

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The highest feasible symmetric projectile trajectory touches the mountain at the top. For symmetric ground-level launch and landing at x=±D, its trajectory with summit height H is y=H(1−x²/D²). Step 2: The hemisphere upper profile is y=√(r²−x²). At the minimum range needed to clear it, the trajectories meet at x=0 with the same height H=r and equal curvature at the top. Step 3: Near x=0 the circle has y≈r−x²/(2r), while the projectile has y≈r−rx²/D². Equate x² coefficients: r/D²=1/(2r),…