Question

The equation of projectiles is
$x = 16x - \binom{x {}^{2} }{4}$
the horizontal range is​

Answer 1

Question :

The equation of the projectile is :

$\sf \: y = 16x - \dfrac{ \: \: {x}^{2} }{4}$

Find the horizontal range.

Solution :

• The horizontal range is 64 m

Given,

$\sf \: y = 16x - \dfrac{ \: \: {x}^{2} }{4}$

Generally,the equation of a projectile is given as :

$\sf y = x.tan( \alpha ) - \dfrac{1}{2} \dfrac{gx {}^{2} }{ {u}^{2}cos {}^{2}( \alpha ) }$

Multiplying and Dividing by sin(∅)

$\longrightarrow \sf y = x.tan( \alpha ) - \dfrac{gx {}^{2}sin( \alpha ) }{[ 2{u}^{2}cos {}^{}( \alpha ) sin( \alpha ) ] \: cos ( \alpha )}$

We know that,

$\sf R = \dfrac{2u^2sin(\alpha)cos(\alpha)}{g}$

Now,

$\longrightarrow \sf \: y = x.tan( \alpha ) - \dfrac{ {x}^{2} tan( \alpha )}{R} \\ \\ \longrightarrow \sf \: y = x.tan( \alpha ) \bigg(1 - \dfrac{x}{R} \bigg) - - - - - - - - - (1)$

Further,

$\sf \: y = 16x - \dfrac{ \: \: {x}^{2} }{4} \\ \\ \implies \: \sf \: y = 16x \bigg(1 - \dfrac{x}{64} \bigg) - - - - - - - (2)$

Comparing equations (1) and (2),we get :

$\boxed{ \boxed{ \sf R = 64 \: units}}$