Two particles are projected simultancously with the samc...

Two particles are projected simultancously with the samc speed $v$ in the same vertical planc with angles of clevation $\theta$ and $2 \theta$, wherc $\theta<45^{\circ}$. At what time will their velocities be parallel?

Key Concepts

Projectile Motion

This concept refers to the motion of an object that is launched into the air and influenced only by gravity (neglecting air resistance). In projectile motion, the overall motion can be analyzed by separately considering the horizontal motion, which occurs at a constant velocity, and the vertical motion, which is subject to the constant acceleration due to gravity.

Decomposition of Velocity Vectors

In analyzing motion in a plane, the velocity vector is broken down into horizontal and vertical components. The horizontal component remains constant in projectile motion, while the vertical component changes linearly with time due to gravity. This decomposition is critical for understanding how each component contributes to the overall motion of a projectile.

Conditions for Parallel Vectors

Two vectors are parallel if they have the same or exact opposite direction, which mathematically means that the ratio of their corresponding components (for example, the vertical to horizontal components) must be equal. In the context of projectile motion, setting the ratios of the vertical to horizontal velocity components equal for different projectiles can help determine the specific time at which their velocities are parallel.

Kinematics Under Constant Acceleration

This concept involves the use of equations that describe the motion of objects under a constant acceleration, such as gravity. These kinematic equations allow us to express velocity components as functions of time, enabling the analysis of the projectile’s motion at any given moment. They are essential for determining when certain conditions—like the alignment of velocity vectors—are met.

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