A particle is projected from a point O with a velocity u in...

A particle is projected from a point $O$ with a velocity $u$ in a direction making an angle $\alpha$ upward with the horizontal. After some time at point $P$ it is moving at right angle with its initial direction of projection. The time of flight from $O$ to $P$ is (a) $\frac{u \sin \alpha}{g}$ (b) $\frac{u \operatorname{cosec} \alpha}{g}$ (c) $\frac{u \tan \alpha}{g}$ (d) $\frac{u \sec \alpha}{g}$

A particle is projected from a point $O$ with a velocity $u$ in a direction making an angle $\alpha$ upward with the horizontal. After some time at point $P$ it is moving at right angle with its initial direction of projection. The time of flight from $O$ to $P$ is
(a) $\frac{u \sin \alpha}{g}$
(b) $\frac{u \operatorname{cosec} \alpha}{g}$
(c) $\frac{u \tan \alpha}{g}$
(d) $\frac{u \sec \alpha}{g}$

Key Concepts

Projectile Motion

Projectile motion is the study of objects launched into the air and moving under the influence of gravity. This topic involves analyzing the path (trajectory) of a moving object when it is subjected to gravitational acceleration, with the motion typically decomposed into horizontal and vertical components.

Decomposition of Initial Velocity

The initial velocity of a projectile is resolved into horizontal and vertical components using trigonometric functions. This decomposition is crucial because the horizontal component (u cos ?) remains constant (in the absence of air resistance) while the vertical component (u sin ?) is affected by gravitational acceleration, enabling the determination of various attributes of the motion.

Time of Flight in Projectile Motion

Time of flight refers to the total duration a projectile remains in the air. Calculating this time generally involves analyzing the vertical displacement and acceleration due to gravity. Depending on specific conditions, such as when the velocity vector makes a given angle with the initial direction, particular formulas can be derived to specify portions of the motion.

Trigonometric Relationships in Kinematics

Trigonometry is essential in kinematics for relating angles to the magnitudes of vector components. Functions like sine, cosine, and tangent, along with their reciprocal functions, help translate between the directional properties of velocity and the scalar quantities needed for calculating time, displacement, and other parameters in projectile motion.

Vector Angles and Relative Directions

Understanding the angle between vectors is key when analyzing changes in the direction of motion in projectile problems. This concept involves using geometry to determine when and how the projectile's velocity vector deviates from its original direction, which is critical in problems that set constraints based on specific angular relationships.

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