Question

A ball is thrown with a velocity $u$ making an angle '$\theta$' with the horizontal. Its velocity vector is normal to initial velocity vector $u$ after a time interval of

• A. $\frac{u\sin \theta }{g}$
• B. $\frac{u}{g\cos \theta }$
• C. $\frac{u}{g\sin \theta }$
• D. $\frac{4\cos \theta }{g}$

Answer 1

The correct option is C $\frac{u}{g\sin \theta }$

Initial velocity, u
$u=\underset{}{u\cos \theta \hat{i}}+u\sin \theta \hat{j}$
$u_x=$ constant

$u_x\left(t\right)=u\cos \theta$
$u_y\left(t\right)=u\sin \theta -gt$
So, $u\left(t\right)=u\cos \theta \hat{i}+(u\sin \theta -gt)\hat{j}$

Now, for $u\left(t\right)$ to be perpendicular to $u$, dot product should be $0$.
$\overrightarrow{u\left(t\right)}\cdot \overrightarrow{u}=0$

$(u\cos \theta \hat{i}+(u\sin \theta -gt\left)\hat{j}\right)\cdot (u\cos \theta \hat{i}+u\sin \theta \hat{j})=0$

$\Rightarrow u^2{\cos }^2\theta +u^2{\sin }^2\theta -u\sin \theta gt=0$

$t=\frac{u}{g\sin \theta }$