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

A particle is projected from point $O$ with velocity $u$ in a direction making an angle $\alpha$ with the horizontal At any instant its position is at point $P$ at right angles to the initial direction of projection. Its velocity at point $P$ is (a) $u \tan \alpha$ (b) $u \cot \alpha$ (c) $u \operatorname{cosec} \alpha$ (d) $u \sec \alpha$

A particle is projected from point $O$ with velocity $u$ in a direction making an angle $\alpha$ with the horizontal At any instant its position is at point $P$ at right angles to the initial direction of projection. Its velocity at point $P$ is
(a) $u \tan \alpha$
(b) $u \cot \alpha$
(c) $u \operatorname{cosec} \alpha$
(d) $u \sec \alpha$

Key Concepts

Projectile Motion

Projectile motion describes the motion of an object thrown into the air, which moves under the influence of gravity. In such problems, the motion is typically broken into horizontal and vertical components where the horizontal component remains constant (in the absence of air resistance) and the vertical component experiences constant acceleration due to gravity.

Vector Decomposition

Breaking the initial velocity or displacement into horizontal and vertical components is a key technique in projectile motion. This decomposition allows for separate analysis of the motion in each direction, making it easier to apply the equations of motion individually to the horizontal and vertical parts.

Dot Product and Perpendicularity

The dot product of two vectors enables determining the angle between them; specifically, if the dot product is zero, the vectors are perpendicular. This principle is used to establish conditions, such as when the instantaneous velocity or position is at a right angle to a given direction, by setting the dot product of the relevant vectors to zero.

Instantaneous Velocity in Projectile Motion

Determining the instantaneous velocity at a specific moment involves calculating its horizontal and vertical components at that time. Once the time is found from a specified condition—such as the velocity vector being perpendicular to the initial velocity—these components can be recalculated and combined to yield the overall speed.

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