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the equation of projectile is y = ax - bx2 . its horizontal range is?

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Everest2017idn
y = a x - b x^2
Range is a/b

y = tan Ф x -  g x²  / 2 u² cos² Ф
tan Ф = a        -  equation 1
b = g / 2u² cos² Ф          so  u² cos² Ф = g /2b    - equation 2

R = u cos Ф * 2 * u sin Ф / g  = 2/g  sinФ  u² cos Ф
     = 2 /g   tan Ф   u² cos² Ф            by using equation 1  and equation 2
     = (2 /g )  a  (g / 2b ) = a / b

Answer: Range, [tex]R =\frac{a}{b}[/tex]

Explanation:

The equation of trajectory is:

[tex]y = x tan \theta (1-\frac{x}{R})[/tex]

Where, [tex]\theta[/tex] is the angle of projectile, R is the horizontal range.

The equation of projectile is:

y = ax-bx²

[tex]\Rightarrow y = ax(1-\frac{b}{a}x)[/tex]

On comparing:

[tex]tan \theta = a[/tex]

[tex]R = \frac{a}{b}[/tex]

Hence, the horizontal range is [tex]R =\frac{a}{b}[/tex]

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