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Question #78241

Difficulty: 5/10 | 🟢 Free

The Question

A particle is projected from the surface of a uniform solid spherical planet of mass $M$ and radius $R$ . The projection speed is $u = \sqrt{\frac{GM}{R}}$ (where $G$ is the universal gravitational constant) and the angle of projection is $\theta = 45^\circ$ with respect to the local horizontal. Assuming the planet does not rotate and has no atmosphere, what is the true horizontal range of the projectile (defined as the distance measured along the surface of the planet between the launch point and the impact point)?

Options

A.
$\frac{\pi R}{2}$
B.
R
C.
$\frac{\pi R}{4}$
D.
$\sqrt{2} R$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Horizontal Range Invariance|Gravitational Field Variation (Depth vs Height)|Conservative Field Path Independence|Integration

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

The core trap tests the misapplication of standard projectile kinematics ( $R = u^2\sin(2\theta) / g$ ) in a regime where the gravitational field is non-uniform and the planet's curvature is significant.

Students who blindly apply the flat-Earth formula will calculate $R = (gR \cdot \sin 90^\circ)/g = R$ , failing to recognize that a projection speed comparable to orbital velocity necessitates a central-force (Keplerian) framework.

The problem forces the reconciliation of energy and angular momentum conservation to find the true angular displacement, punishing the assumption of a parabolic trajectory with an elliptical one.

Additionally, it distinguishes between the arc length along the surface (the true range) and the straight-line chord displacement ( $\sqrt{2}R$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Horizontal Range Invariance
Difficulty	5
Cognitive Load	6
Target Solve Time	140 sec
Access	Free