Question

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 $\left(\right.30 \, m, \, 40 \, m\left.\right)$ ? Take $g \, = \, 10 \, m \, s^{- 2}$

• A. $60ms^{- 1}$
• B. $30 \, m \, s^{- 1}$
• C. $50ms^{- 1}$
• D. $40ms^{- 1}$

Answer 1

Answer: (B)

Using equation of trajectory for a projectile motion :
$y=xtan \theta - \frac{g x^{2}}{2 u^{2}} \left(1 + \left(tan\right)^{2} ⁡ \theta \right)$
For the projectile to pass through (30 m, 40 m) :
$40=30 \tan \alpha-\frac{\mathrm{g}(30)^2}{2 \mathrm{u}^2}\left(1+\tan ^2 \alpha\right)$
or $900\left(tan\right)^{2}\alpha -\left(\right.6u^{2}tan\alpha \left.\right)+\left(\right.900+8u^{2}\left.\right)=0$
For real value of $\alpha $ . The discriminant of a quadratic equation $b^{2}-4ac\geq 0$
$\left(6 u^{2}\right)^{2} \geq 3600\left(900+8 u^{2}\right)$
or $\quad\left(u^{4}-800 u^{2}\right) \geq 9,00,00$
or $\left(u^{2}-400\right)^{2} \geq(2,50,000)$
or $u^{2}-400 \geq 500$
or $u^{2} \geq 900$ or $u \geq 30 m s ^{-1}$