A particle is projected from the ground with an initial speed of u at an angle of projection \[\theta \]. The average velocity of the particle between its time of projection and time it reaches highest point of trajectory is:
A. \[\dfrac{u}{2}\sqrt{1+2{{\cos }^{2}}\theta }\]
B. \[\dfrac{u}{2}\sqrt{1+2{{\sin }^{2}}\theta }\]
C. \[\dfrac{u}{2}\sqrt{1+3{{\cos }^{2}}\theta }\]
D. \[u\cos \theta \]

Hint: Projectile motion is a type of motion experienced by an object or molecule (a projectile) that is anticipated close to the Earth's surface and moves along a bended way under the activity of gravity just (specifically, the impacts of air opposition are thought to be insignificant).

Complete step-by-step answer:
The correct option is C

The investigation of such motions is called ballistics, and such a trajectory is a ballistic trajectory.
Height of the projectile at its highest point is

Horizontal distance covered till this point is \[h=\dfrac{{{u}^{2}}+{{\sin }^{2}}\theta }{2g}\]

Time taken to reach highest point is \[t=\dfrac{T}{2}=\dfrac{u\sin \theta }{g}\]

\[t=\sqrt{{{h}^{2}}+{{x}^{2}}}=\dfrac{{{u}^{2}}\sin \theta }{g}(\sqrt{1}+3{{\cos }^{2}}\theta )\]

Hence, displacement is \[x=\dfrac{R}{2}=\dfrac{{{u}^{2}}\sin 2\theta }{2g}=\dfrac{{{u}^{2}}\sin \theta \cos \theta }{2g}\]

Average velocity \[v=\dfrac{d}{t}=\dfrac{u}{2}\sqrt{1+3{{\cos }^{2}}\theta )}\]

The main force of essentialness that follows up on the object is gravity, which acts descending, along these lines conferring to the object a descending speed up. Due to the object's latency, no outside flat force is expected to keep up the level velocity segment of the object. Considering different forces, for example, erosion from streamlined drag or inward drive, like, in a rocket, requires extra examination.

A ballistic rocket is a rocket just guided during the moderately short starting powered period of flight, and whose ensuing course is administered by the laws of old-style mechanics.

Ballistics ("to toss") is the study of mechanics that manages the flight, conduct, and impacts of projectiles, particularly shots, unguided bombs, rockets, or something like that; the science or craft of structuring and quickening projectiles in order to accomplish an ideal exhibition.

Note: In projectile motion, horizontal motion and the vertical motion are free of one another; that is, neither one of the motions influences the other. This is the guideline of compound motion set up and utilized by him to demonstrate the explanatory type of projectile motion. The horizontal and vertical segments of a projectile's velocity are autonomous of one another.