From a point at a height h above the oround, a particle A is...

From a point at a height $h$ above the oround, a particle $A$ is projected with a velocity $v$ in an upward direction making an angle $\theta$ with the horizontal. $\Lambda$ nother particle $B$ is projected from the same point with the same velocity $v$ but, in a direction directly opposite to $A$. Show that the two particles hit the ground al a distanec $\frac{2 v}{g} \cos \theta \sqrt{\left(v^{2} \sin ^{2} \theta+2 g h\right)}$ apart.

Key Concepts

Projectile Motion

This concept describes the motion of an object under the influence of gravity, where the only acceleration acting on the object is due to gravity. It assumes that air resistance is negligible and that the horizontal and vertical motions can be analyzed independently, leading to a parabolic trajectory.

Decomposition of Velocity

In analyzing projectile motion, the initial velocity is broken down into two orthogonal components: one horizontal and one vertical. This decomposition, typically using trigonometric functions like sine and cosine, allows the separate treatment of constant horizontal motion and uniformly accelerated vertical motion.

Time of Flight

The time a projectile remains in the air, or time of flight, is determined by examining its vertical motion under gravity. Especially when launched from an elevated position, this involves solving a quadratic equation obtained from the kinematic equation for vertical displacement to find the time until the projectile reaches the ground.

Horizontal Displacement (Range) Calculation

Since the horizontal velocity component remains constant in projectile motion (ignoring air resistance), the horizontal displacement or range is found by multiplying this constant horizontal speed by the time of flight. When multiple projectiles are involved, calculating their respective horizontal ranges allows for determining distances between their landing points.

Quadratic Equations in Kinematics

Projectile problems, particularly those involving non-zero initial heights, lead to quadratic equations in time when setting up the vertical displacement equation. Solving these quadratic equations is essential to finding the time of flight which then influences other aspects of the projectile’s motion, such as range.

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