Taking X -axis along horizontal and Y -axis along vertical, with what minimum speed must a particle be projected from the origin so that it is able to pass through a given point ( 30 m , 40 m ) ? Take

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Taking X -axis along horizontal and Y -axis along vertical, with what minimum speed must a particle be projected from the origin so that it is able to pass through a given point ( 30 m , 40 m ) ? Take g = 10 m s - 2

TestHub · Accepted Answer

Using equation of trajectory for a projectile motion :

$y = x \tan θ - \frac{g x^{2}}{2 u^{2}} (1 + \tan^{2} θ)$

For the projectile to pass through (30 m, 40 m) :

$40 = 30 tan α - \frac{{g (30)}^{2}}{{2u}^{2}} (1 + {tan}^{2} α)$

or $900 \tan^{2} α - (6 u^{2} \tan α) + (900 + 8 u^{2}) = 0$

For real value of $α$ . The discriminant of a quadratic equation $b^{2} - 4 a c \geq 0$

$(6 u^{2})^{2} \geq 3600 (900 + 8 u^{2})$

or $(u^{4} - 800 u^{2}) \geq 9, 00, 00$

or $(u^{2} - 400)^{2} \geq (2, 50, 000)$

or $u^{2} - 400 \geq 500$

or $u^{2} \geq 900$ or $u \geq 30 m s^{- 1}$

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