A projectile is projected from ground at a speed of $30{\mathrm{ms}}^{-1}$ at an angle of $30°$ with the horizontal. At what instant the vertical displacement of projectile is $5\mathrm{m}$ ? Interpret the answer. [Take $g=10{\mathrm{ms}}^{-2}$ ]

Step 1: Given data

The initial velocity of a projectile, $u=30m{s}^{-1}$

The angle of projection, $\theta =30°$

The vertical displacement of a projectile, $S_y=5m$

Step 2: Find the instant when the vertical displacement of the given projectile is $\mathbf{5}\mathbf{}\mathbf{m}$

It is given that the vertical displacement of a projectile is $S_y=5m$

Also, the second equation of motion for $Y-axis$ is ${\mathit{S}}_{\mathbf{y}}\mathbf{=}\mathbf{}{\mathit{u}}_{\mathbf{y}}\mathit{t}\mathbf{+}\frac{\mathbf{1}}{\mathbf{2}}{\mathit{a}}_{\mathbf{y}}{\mathit{t}}^{\mathbf{2}}$.

Here, $u_y$ is the vertical component of the initial velocity and $a_y$ is the acceleration acting on the $Y-axis$ and $t$ is the time taken to complete the displacement $S_y$.

$\therefore S_y=u_{y}t+\frac{1}{2}{a}_y{t}^2$

$5=u\sin \theta \times t+\frac{1}{2}(-g)t^2\left[\because u_y=u\sin \theta \right]$

$$\begin{gathered} 5=30\sin 30\times t-\frac{1}{2}\times 10t^2 \\ 5=30\times \frac{1}{2}\times t-5t^2 \\ 5=15t-5t^2 \\ 1=3t-t^2 \\ {t}^2-3t+1=0 \\ {t}^2+{\left(\frac{3}{2}\right)}^2-{\left(\frac{3}{2}\right)}^2-3t+1=0 \\ {\left(t-\frac{3}{2}\right)}^2-\frac{9}{4}+1=0\left[\because {(a-b)}^2=a^2+b^2-2ab\right] \\ {\left(t-\frac{3}{2}\right)}^2=\frac{9}{4}-1 \\ {\left(t-\frac{3}{2}\right)}^2=\frac{9-4}{4} \\ {\left(t-\frac{3}{2}\right)}^2=\frac{5}{4} \\ t-\frac{3}{2}=\sqrt{\frac{5}{4}} \\ t-\frac{3}{2}=\frac{\sqrt{5}}{2}andt-\frac{3}{2}=-\frac{\sqrt{5}}{2} \\ t=\frac{\sqrt{5}}{2}+\frac{3}{2}andt=-\frac{\sqrt{5}}{2}+\frac{3}{2} \\ t=\frac{3+\sqrt{5}}{2}andt=\frac{3-\sqrt{5}}{2} \\ t=2.62sandt=0.38s \end{gathered}$$

Hence, the instants when the vertical displacement of the given projectile is $\mathbf{5}\mathbf{}\mathbf{m}$ are $\mathbf{0}\mathbf{.}\mathbf{38}\mathbf{}\mathit{s}$ and $\mathbf{2}\mathbf{.}\mathbf{62}\mathbf{}\mathit{s}$.