How Krotov's Challenging Projectile Problem Becomes An Easy One | Krotov 1.10

In this Physics video in Hindi for the chapter : "Motion in a Plane" of Class 11, we discussed problem no. 1.10 from the book 'Aptitude Test Problems in Physics' by S.S. Krotov.

The Problem states :-
A cannon fires from under a shelter inclined at an angle 𝜶 to the horizontal. The cannon is at a distance 𝒍 from the base of the shelter. The initial velocity of the shell is 𝒗₀, and its trajectory lies in the plane of the figure. Determine the maximum range 𝑳ₘₐₓ of the shell.


This problem demands a deep conceptual understanding of two-dimensional motion and careful visualisation of the geometry involved, making it an enriching exercise for IIT-JEE aspirants.

The question states that the cannon fires from point A under a sloping roof inclined at a certain angle, and the shell must exit through the open side without touching the shelter. We analyse how the initial velocity direction must be chosen to just clear the roof while still achieving the greatest possible horizontal distance. In the explanation, we break down the motion into horizontal and vertical components and track the projectile’s path relative to the shelter’s geometry. This allows us to identify the limiting condition for escape and the orientation that produces the maximum possible range.

To solve the problem, we first interpret the geometry of the shelter and determine the boundaries within which the projectile must remain. Then we analyse the motion using the essential principles of two-dimensional kinematics, focusing on the projectile’s initial direction and the constraints imposed by the roof. By examining the physically allowed region and the conditions for safe exit, we construct the logic required to identify the direction that leads to the maximum achievable range on level ground.

Definition 1: Projectile Motion.
Projectile motion is the two-dimensional motion of an object under the influence of gravity alone, characterised by simultaneous horizontal uniform motion and vertical accelerated motion. It is a fundamental concept used throughout this question to describe the shell’s trajectory.

Definition 2: Range of a Projectile.
The range of a projectile is the horizontal distance travelled by the object from the point of projection to the point where it lands. In this question, the range must be maximised within the restrictions created by the shelter’s geometry.

Theorem 1: Independence of Horizontal and Vertical Motions.
This theorem states that the horizontal and vertical motions of a projectile are independent of each other except for sharing the same time of flight. This principle allows us to separately analyse the horizontal distance and the vertical height constraints.

Theorem 2: Condition for Optimal Projection in Constrained Motion.
When a projectile is launched inside a restricted environment, the optimal direction occurs when the trajectory just grazes the limiting boundary. In this problem, the optimal trajectory is the one that just clears the roof of the inclined shelter, ensuring maximum possible range under the constraint.

Throughout the explanation, the chapter "Motion in a Plane" appears repeatedly as the conceptual backbone of the discussion. The techniques used are particularly valuable for students aiming for IIT-JEE Advanced, where non-standard projectile motion problems are frequently encountered. The book by S.S. Krotov is known for its elegant and challenging problems, and this question reflects that depth by combining geometry with kinematic reasoning.

By the end of the video, you will gain a thorough understanding of how to apply projectile motion theory in a constrained environment, how to evaluate limiting conditions, and how to reason your way toward maximising the horizontal range. This approach is essential for excelling in advanced problem-solving for IIT-JEE Advanced, Olympiads, and other competitive examinations.

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