5.2 Features of projectile motion  (Page 3/7)

$$\begin{array}{l}x=u_{x}t\\ \Rightarrow R=u_{x}T=u\cos \theta xT\end{array}$$

where “T” is the time of flight. Putting expression for the time of flight,

$$\begin{array}{l}\Rightarrow R=\frac{u\cos \theta x2u\sin \theta }{g}\end{array}$$

Horizontal range, like time of flight and maximum height, is greater for greater projection speed.

For projection above ground surface, the range of the angle of projection with respect to horizontal direction, θ, is 0° ≤ θ ≤ 90° and the corresponding range of 2θ is 0° ≤ 2θ ≤ 180°. The “sin 2θ”, as appearing in the numerator for the expression of the horizontal range, is an increasing function for 0° ≤ θ ≤ 45° and a decreasing function for 45° ≤ θ ≤ 90° . For θ = 45°, sin 2θ = sin 90° = 1 (maximum).

In the nutshell, the range,R, increases with increasing angle of projection for 0° ≤ θ<45°; the range,R, is maximum when θ = 45°; the range,R, decreases with increasing angle of projection for 45°<θ ≤ 90°.

The maximum horizontal range for a given projection velocity is obtained for θ = 45° as :

$$\begin{array}{l}\Rightarrow R=\frac{u^2}{g}\end{array}$$

There is an interesting aspect of “sin 2θ” function that its value repeats for component angle i.e (90° – θ). For, any value of θ,

$$\begin{array}{l}\sin 2({90}^0-\theta )=\sin ({180}^0-2\theta )=\sin 2\theta \end{array}$$

It means that the range of the projectile with a given initial velocity is same for a pair of projection angles θ and 90° – θ. For example, if the range of the projectile with a given initial velocity is 30 m for an angle of projection, θ = 15°, then the range for angle of projection, θ = 90° - 15° = 75° is also 30m.

Equation of projectile motion and range of projectile

Equation of projectile motion renders to few additional forms in terms of characteristic features of projectile motion. One such relation incorporates range of projectile (R) in the expression. The equation of projectile motion is :

$$y=x\tan \theta -\frac{g{x}^2}{2u^2{\cos }^2\theta }$$

The range of projectile is :

$$R=\frac{u^2\sin 2\theta }{g}=\frac{u^{2}2\sin \theta \cos \theta }{g}$$

Solving for $u^2$ ,

$$\Rightarrow u^2=\frac{Rg}{2\sin \theta \cos \theta }$$

Substituting in the equation of motion, we have :

$$\Rightarrow y=x\tan \theta -\frac{2\sin \theta \cos \theta g{x}^2}{2Rg\cos {}^2\theta }$$ $$\Rightarrow y=x\tan \theta -\frac{\tan \theta x^2}{R}$$ $$\Rightarrow y=x\tan \theta \left(1-\frac{x}{R}\right)$$

It is a relatively simplified form of equation of projectile motion. Further, we note that a new variable “R” is introduced in place of “u”.