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Question #77657

Difficulty: 5/10 | 🟢 Free

The Question

A small particle is projected horizontally with an initial speed $v_0 \text{ (m/s)}$ from a point $O$ on a fixed, long inclined plane. The inclined plane makes an angle $\theta \text{ (rad)}$ with the horizontal. The coefficient of restitution between the particle and the incline is $e$ (dimensionless). Assume the surface of the incline is perfectly smooth. Find the total distance $L \text{ (m)}$ measured along the incline from the projection point $O$ to the point where the particle strikes the incline for the second time. Acceleration due to gravity is $g \text{ (m/s}^2\text{)}$ .

Options

A.
$\frac{2 v_0^2 \sin\theta}{g \cos^2\theta} \left[ 1 + e + e(1+e)\sin^2\theta \right]$
B.
$\frac{2 v_0 \sin\theta}{g \cos^2\theta} \left[ 1 + e + e(1+e)\sin^2\theta \right]$
C.
$\frac{2 v_0^2 \sin\theta}{g \cos^2\theta} \left[ 1 + e + e(1-e)\cos^2\theta \right]$
D.
$\frac{v_0^2 \sin\theta}{g \cos^2\theta} \left[ 1 + e^2 + e(1+e)\sin^2\theta \right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Projectile Path Equation|Impulse-Momentum Theorem|Vector Decomposition

Why This Question Is Tricky

Examiner Levers

Dimensional Analysis

Tests if the student verifies units/dimensions before selecting.

Difficulty Boost : +2

Common Mistakes

This question tests the student's ability to correctly apply coordinate transformations and the Impulse-Momentum theorem in a non-standard projectile scenario.

The primary trap lies in assuming the horizontal velocity component (relative to the ground) remains constant, failing to realize that gravity has a component along the incline ( $g \sin\theta$ ) which continuously accelerates the particle in the $x'$ -direction.

Furthermore, students often incorrectly reset the velocity vector after collision by applying the coefficient of restitution $e$ to the net velocity or the horizontal ground component, rather than strictly to the component normal to the incline.

The inclusion of a dimensionally incorrect option (B) also penalizes students who skip basic dimensional analysis checks.

Quick Facts About This Question

Question Information

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