Question

A particle is projected with a velocity $v$ such that its range on the horizontal plane is twice the greatest height attained by it. The range of the projectile is (where $g$ is acceleration due to gravity)

• A. $\cos 25^{\circ}$ to $\cos 50^{\circ}$
• B. $\frac{2 v^{2}}{3 g}$
• C. $\frac{4 v^{2}}{5 \,g}$
• D. $\frac{v^{2}}{2 g}$

Answer 1

The Correct Option is C

Given, Range, \(R = 2 H\) But \(R =4 H \cot \theta\) \(\Rightarrow \cot \theta=\frac{1}{2}\) From triangle we can say that, \(\sin \theta=\frac{2}{\sqrt{5}}\) and \(\cos \theta=\frac{1}{\sqrt{5}}\) So, the range of the projectile \(R =\frac{2 v ^{2} \sin \theta \cos \theta}{ g }=\frac{2 v ^{2}}{ g } \times \frac{2}{\sqrt{5}} \times \frac{1}{\sqrt{5}}\) \(=\frac{4 v ^{2}}{5 g }\) 

A projectile is anything or everything launched into space on which only gravity acts. The sole fundamental operating force on a projectile is gravity. It nevertheless experiences other forces, only to a far lesser extent than gravity, which does not imply that they do not affect it. A projectile's trajectory is the path it takes while travelling. An example of a projectile is a batted or tossed tennis ball.

Generally speaking, there are three different sorts of projectiles:

a projectile that is allowed to fall freely from a significant height.

Straight-up projectile that is launched.

at an angle to the horizontal, launch upward.

A particle that is flung obliquely towards the earth's surface travels along a curved route with constant acceleration that is pointed in the direction of the planet's centre. Such a particle's motion is referred to as projectile motion, and its route is known as a projectile.

In Projectile Motion, there are two separate, concurrent rectilinear motions: 

Along the x-axis: Uniform Velocity which is repsonsible for the horizontal (forward) motion of the particle.

Along the y-axis: Uniform Acceleration which is repsonsible for the vertical (downward) motion of the particle.

Acceleration of a particle in horizontal and vertical projectile motion is:

Gravitational acceleration (g) moves downward and vertically. 

There is no acceleration of the bullet in its horizontal path, as evidenced by the projectile's steady horizontal velocity.