A particle is projected with a velocity u, at an angle α, with the horizontal. Time at which its

step 1 Hi, here in this given problem there is a projectile which is projected with an initial speed u

A particle is projected with a velocity u, at an angle α,...

A particle is projected with a velocity $u$, at an angle $\alpha$, with the horizontal. Time at which its vertical component of velocity becomes half of its net speed at the highest point will be (A) $\frac{u}{2 g}$ (B) $\frac{u}{2 g}(\sin \alpha-\cos \alpha)$ (C) $\frac{u}{2 g}(2 \cos \alpha-\sin \alpha)$ (D) $\frac{u}{2 g}(2 \sin \alpha-\cos \alpha)$

A particle is projected with a velocity $u$, at an angle $\alpha$, with the horizontal. Time at which its vertical component of velocity becomes half of its net speed at the highest point will be
(A) $\frac{u}{2 g}$
(B) $\frac{u}{2 g}(\sin \alpha-\cos \alpha)$
(C) $\frac{u}{2 g}(2 \cos \alpha-\sin \alpha)$
(D) $\frac{u}{2 g}(2 \sin \alpha-\cos \alpha)$

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is thrown or projected into the air and is subject only to the acceleration of gravity. This type of motion has both horizontal and vertical components, with the horizontal motion being uniform (constant velocity) and the vertical motion being uniformly accelerated (due to gravity).

Decomposition of Velocity into Components

In projectile motion, the initial velocity is typically resolved into two components: horizontal and vertical. The horizontal component, found using the cosine of the launch angle, remains constant throughout the flight (ignoring air resistance), while the vertical component, found using the sine of the launch angle, is affected by gravity, which causes it to decrease over time.

Kinematic Equations under Constant Acceleration

Kinematic equations describe the motion of objects under constant acceleration. For projectile motion, these equations are applied separately to the horizontal and vertical components of velocity and displacement. The vertical motion is particularly notable because it involves a constant acceleration, namely gravity, which allows calculation of quantities like time, height, and velocity at any given point in time.

Trigonometric Relationships in Kinematics

Trigonometric relationships are essential in resolving the initial velocity into horizontal and vertical components. By relating the sine and cosine of the launch angle to the vertical and horizontal components, respectively, these relationships facilitate the analysis of projectile motion by allowing complex motion to be broken down into simpler, one-dimensional motions.

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