A particle of mass m is projected with velocity v making an...

A particle of mass $m$ is projected with velocity $v$ making an angle of $45^{\circ}$ with the horizontal. The magnitude of the angular momentum of the particle about the point of projection when the particle is at its maximum height is (where $g$ = acceleration due to gravity) (a) Zero (b) $m v^{3} /(4 \sqrt{2} g)$ (c) $m v^{3} /(\sqrt{2} g)$ (d) $m v^{2} / 2 g$

A particle of mass $m$ is projected with velocity $v$ making an angle of $45^{\circ}$ with the horizontal. The magnitude of the angular momentum of the particle about the point of projection when the particle is at its maximum height is (where $g$ = acceleration due to gravity)
(a) Zero
(b) $m v^{3} /(4 \sqrt{2} g)$
(c) $m v^{3} /(\sqrt{2} g)$
(d) $m v^{2} / 2 g$

Key Concepts

Velocity Components at Maximum Height

At the maximum height of a projectile's trajectory, the vertical component of velocity becomes zero due to the effect of gravity, leaving only the horizontal component. This simplification is crucial when calculating the angular momentum at that point, as it directly affects how the velocity contributes to the cross product with the position vector.

Cross Product in Angular Momentum Calculation

The calculation of angular momentum using the cross product involves determining the perpendicular component of the momentum relative to the position vector. This mathematical operation inherently accounts for the moment arm—the distance at which the momentum is applied—thus providing both the magnitude and direction of the angular momentum.

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject only to acceleration due to gravity. Key parameters include the initial velocity, launch angle, time of flight, maximum height, and horizontal range. Understanding the basic kinematics allows for the analysis of various aspects such as position, velocity, and acceleration at any point along the trajectory.

Angular Momentum

Angular momentum is a physical quantity that represents the rotational analogue of linear momentum. It is defined for a particle as the cross product of its position vector relative to a chosen origin and its linear momentum. The magnitude and direction of angular momentum depend on both the distance from the point of rotation and the component of velocity perpendicular to that distance.

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