A particle when projected in vertical planc moves along the...

A particle when projected in vertical planc moves along the fixed smooth surfacc with initial velocity $20 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$, so that its normal reaction on the surlace remains isro throughout the motion. Then find the slope of the tangent to the surface at height $5 \mathrm{~m}$ from the point of projoction $A$.

Key Concepts

Projectile Motion

This concept deals with the motion of objects moving under the influence of gravity alone, with their initial velocity decomposed into horizontal and vertical components. Using kinematic equations, one can model the object’s path, time of flight, maximum height, and range, all of which are fundamental to understanding the trajectory in a uniform gravitational field.

Free-Fall Trajectory

A free-fall trajectory describes the parabolic path followed by an object when only gravity acts on it. In this concept, air resistance is neglected, and the object’s motion is fully characterized by the initial velocity and the constant acceleration due to gravity, resulting in predictable displacement and velocity at any given time.

Constraint Forces and Contact Conditions

When an object moves along a fixed smooth surface, the surface may exert a normal (perpendicular) force on the object. However, if the normal reaction remains zero throughout the motion, the object is not being pushed against the surface; instead, the surface must be tangent to the object's free-fall trajectory at all points. This concept emphasizes the relationship between the geometric path of motion and the constraint (or lack thereof) imposed by the surface.

Tangent to a Curve and Slope Determination

The tangent to a curve at any given point represents the instantaneous direction of the curve, and its slope is calculated as the derivative of the function describing the path. Identifying the slope of the tangent at a specific point is crucial for determining the direction of motion or the inclination of the surface at that point.

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