Question

A projectile is fired from the surface of the Earth of radius $R$ with a velocity $\eta v_{e}$ , where $v_{e}$ is the escape velocity and $\eta < 1$ . Neglecting the air resistance, the orbital velocity of the projectile is

• A. $v_{e}\sqrt{1 - \eta^{2}}$
• B. $v_{e}\sqrt{\frac{\eta^{2}}{5}}$
• C. $\frac{2}{5}v_{e}\sqrt{\eta}$
• D. $\frac{2 \eta}{5}v_{e}$

Answer 1

The Correct Option is A

Velocity at the surface $=v_{s}=\etav_{e}$ Velocity in orbit $=v_{0}$ By conversation of mechanical energy, $ \, \frac{1}{2}mv_{s}^{2}+\left(- \frac{G M m}{R}\right)=\frac{1}{2}mv_{0}^{2}-\frac{G M m}{r}\ldots \left(1\right)$ By Newton's law, $\frac{G M m}{r^{2}}=\frac{m v_{0}^{2}}{r}\ldots \left(2\right)$ Solving equations (1) and (2) $v_{0}=v_{e}\sqrt{1 - \eta^{2}}$

Answer 2

Ans. A trajectory is a path or track that an object takes as it is launched into the air while being pulled by gravity. This thing is called a projectile, and it forms a parabolic curve as it moves in two dimensions. Real-world sports examples of trajectory include javelin throws, football, soccer, and hockey. The projectile has a starting velocity that propels it forward and maintains its motion; gravity force is the only force acting on it, regardless of the direction of flight. Here, the two motion components are independent of one another.

A projectile is an object set in flight by applying an external force. The projectile may be thrown up vertically or at an angle to the horizontal. It may be dropped from a position of rest. The only force acting on the projectile during its motion along the flight path is the gravitational force and it is in motion due to its own inertia. Examples of projectiles are a ball hit by a bat, a bullet fired from a gun, a shell launched from a launcher, a bomb dropped from a plane, etc. It must be noted that a rocket or a missile cannot be considered a projectile as they are propelled by power. 

The path of flight of a projectile is called its trajectory. It is a parabolic curve. In order to analyze the projectile motion or the trajectory, it is split into two components namely - vertical and horizontal. The projectile moves horizontally under the influence of the horizontal force acting on it and it moves downwards under the effect of the gravitational pull exerted on it.