A projectile is given an initial velocity of ( ^ i + 3 ^ j )ms^ - 1...

Velocity of projectile at time $t$ is given by

v = u - gt\overset{^}{j}

v = \overset{^}{i} + \sqrt{3}\overset{^}{j} - 10t\overset{^}{j}

$∴ r = ∫ vdt$

= ∫ \left(\overset{^}{i} + \sqrt{3}\overset{^}{j} - 10t\overset{^}{j}\right)dt

= t\overset{^}{i} + \sqrt{3}t\overset{^}{j} - 10\dfrac{t2}{2}\overset{^}{j}

= t\overset{^}{i} + \sqrt{3}t\overset{^}{j} - 5t^{2}\overset{^}{j}

r = t\overset{^}{i} + \left(\sqrt{3}t - 5t^{2}\right)\overset{^}{j}

or x\overset{^}{i} + y\overset{^}{j} = t\overset{^}{i} + \left(\sqrt{3}t - 5t^{2}\right)\overset{^}{j}

On comparing, $x = t$ & & & &...(i)

and $y = \sqrt{3}t - 5t^{2}$ & & & &...(ii)

From Eqs (i) and (ii), we get

$y = \sqrt{3}x - 5x^{2}$