You are at a distance R = 1.5 × 10⁶ m from the centre of an unknown planet. You notice that if you throw a ball horizontally, it goes completely around the planet hitting you in the back 90,000 seconds later with exactly the same speed that you originally threw it. If the length of semi major axis of trajectory of ball is 2R, what is the mass of the planet? Express in scientific notation (a × 10ᵇ) kg and fill (b − 3)/a in OMR sheet.
Physics · Mixed Chapter
JEE Mains

Question

You are at a distance R = 1.5 × 10⁶ m from the centre of an unknown planet. You notice that if you throw a ball horizontally, it goes completely around the planet hitting you in the back 90,000 seconds later with exactly the same speed that you originally threw it. If the length of semi major axis of trajectory of ball is 2R, what is the mass of the planet? Express in scientific notation (a × 10ᵇ) kg and fill (b − 3)/a in OMR sheet.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The orbital period obeys Kepler’s third law T²=4π²a_orb³/(GM), with a_orb=2R=3.0×10⁶ m and T=90,000 s. Step 2: M=4π²a_orb³/(GT²). Step 3: Using G=6.67×10⁻¹¹ gives M≈1.97×10²¹ kg≈2×10²¹ kg. Step 4: In the form a×10ᵇ, a≈2 and b=21, so (b−3)/a=(21−3)/2=9.