Physics - Kinematics Question with Solution | TestHub

Velocity of projectile at time $t$ is given by

$v=u-g t \hat{\mathrm{j}}$

$v=\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}-10 t\widehat{ \mathrm{j}}$

$\therefore r=\int v dt$

$=\int \left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}-10 t \hat{\mathrm{j}}\right) dt$

$=t \hat{\mathrm{i}}+\sqrt{3} t \hat{\mathrm{j}}-10\frac{t^2}{2}\hat{\mathrm{j}}$

$=t \hat{\mathrm{i}}+\sqrt{3} t \hat{\mathrm{j}}-5 t^2 \hat{\mathrm{j}}$

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

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

On comparing, $x=t$          ...(i)

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

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

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