When a particle is projected from the surface of earth, it...

When a particle is projected from the surface of earth, it mechanical energy and angular momentum about center of earth at all time is constant (i) A particle of mass $\mathrm{m}$ is projected from the surface of earth with velocity $\mathrm{V}_{0}$ at angle $\theta$ with horizontal suppose $\mathrm{h}$ be the maximum height of particle from surface of earth and $\mathrm{v}$ its speed at that point them $\mathrm{V}$ is (A) $\mathrm{V}_{0} \cos \theta$ $(\mathrm{B})>\mathrm{V}_{0} \cos \theta$ (C) $<\mathrm{V}_{0} \cos \theta$ (D) zero (ii) Maximum height h of the particle is $(\mathrm{A})=\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$ (B) $>\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$ $(\mathrm{C})<\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$ (D) can be greater than or less than $\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$

When a particle is projected from the surface of earth, it mechanical energy and angular momentum about center of earth at all time is constant
(i) A particle of mass $\mathrm{m}$ is projected from the surface of earth with velocity $\mathrm{V}_{0}$ at angle $\theta$ with horizontal suppose $\mathrm{h}$ be the maximum height of particle from surface of earth and $\mathrm{v}$ its speed at that point them $\mathrm{V}$ is
(A) $\mathrm{V}_{0} \cos \theta$
$(\mathrm{B})>\mathrm{V}_{0} \cos \theta$
(C) $<\mathrm{V}_{0} \cos \theta$
(D) zero
(ii) Maximum height h of the particle is
$(\mathrm{A})=\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$
(B) $>\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$
$(\mathrm{C})<\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$
(D) can be greater than or less than $\left[\left(\mathrm{V}_{0}^{2} \sin ^{2} \theta\right) / 2 \mathrm{~g}\right]$

Key Concepts

Limitations of the Constant Gravity Approximation

The common equations for projectile motion, such as the expression for maximum height (h = V0^2 sin^2?/2g) or the constant horizontal speed (V0 cos?), are derived under the assumption that gravitational acceleration g is constant. However, when considering the full central force nature of gravity and its dependence on the radial distance from Earth’s center, these formulas may not perfectly describe the motion, especially for trajectories that reach significant altitudes, thereby affecting both the speed and maximum height compared to the constant-g predictions.

Motion in a Central Gravitational Field

Projectile motion under an inverse?square law gravitational field is fundamentally different from the usual constant–g approximation. When a particle is projected from Earth’s surface, the gravitational force is central and varies with distance. This leads to trajectories that are segments of conic sections, and the standard kinematic formulas (like those for maximum height using constant g) become approximations valid only when the motion is confined very near the Earth's surface.

Conservation of Mechanical Energy

This principle states that in the absence of non?conservative forces, the sum of kinetic and potential energy of a particle remains constant over its motion. In the context of particles moving under gravity (a conservative force), even if the potential energy is expressed in a non?linear or central form, the total mechanical energy is conserved, allowing one to relate speeds at different points in the trajectory by equating the energy at those points.

Conservation of Angular Momentum

In a central force field, such as Earth's gravitational field when treated as emanating from a point mass, there is no external torque about the center. Consequently, a particle’s angular momentum remains constant throughout its motion. This conservation links the particle’s speed and its distance from the center, and it plays a key role in analyzing trajectories, particularly in deducing how the tangential component of velocity changes with the radial distance.

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