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

A particle of mass $m$ is projected with a velocity $v$ making an angle of $45^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about an axis of projection when the particle is at maximum height $h$ is (a) Zero (b) $\frac{m v^{2}}{4 \sqrt{2} g}$ (c) $\frac{m v^{2}}{\sqrt{2} g}$ (d) $m \sqrt{2 g h^{3}}$

A particle of mass $m$ is projected with a velocity $v$ making an angle of $45^{\circ}$ with the horizontal. The magnitude of angular momentum of the projectile about an axis of projection when the particle is at maximum height $h$ is
(a) Zero
(b) $\frac{m v^{2}}{4 \sqrt{2} g}$
(c) $\frac{m v^{2}}{\sqrt{2} g}$
(d) $m \sqrt{2 g h^{3}}$

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Key Concepts

Angular Momentum

Angular momentum is a measure of the rotational motion of a particle or system. It is defined as the cross product of the position vector and the linear momentum vector. This concept is fundamental in determining how a particle's motion contributes to rotation about a given point or axis.

Projectile Motion

Projectile motion describes the trajectory of an object that is launched into the air and moves under the influence of gravity alone. This concept involves analyzing the horizontal and vertical components of motion separately, and is key to understanding the path the object takes, including its maximum height.

Maximum Height in Projectile Motion

At maximum height, the vertical component of the projectile's velocity becomes zero. This particular condition is useful for simplifying calculations in projectile motion problems, such as determining the displacement and analyzing rotational effects like angular momentum at that point.

Vector Cross Product

The vector cross product is an operation between two vectors that results in a third vector perpendicular to the plane containing the initial vectors. Its magnitude, which includes a sine term of the angle between the vectors, is used when calculating angular momentum, making it critical to understanding rotational dynamics.

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