Question

A body is projected with velocity u at an angle theta with the horizontal. The velocity of the body will become perpendicular to the velocity of projection after a time t given by:
(A) $\dfrac{{2u\sin \theta }}{g}$
(B) $\dfrac{{u\sin \theta }}{g}$
(C) $\dfrac{{2u}}{{g\sin \theta }}$
(D) $\dfrac{u}{{g\sin \theta }}$

Answer 1

Hint: The body is projected at an angle. Taking the equation of motion in vertical and horizontal direction will help in solving further.
Formula used: We will start by solving equations in vertical and horizontal direction.
We will also be using the equations of motion.
$\Rightarrow$ $v = u + at$
$\Rightarrow$ ${{\text{v}}^{\text{2}}}{\text{ = }}{{\text{u}}_{}}^{\text{2}}{\text{ + 2gs}}$

Complete step by step answer:
Here, we already know that there will be no change in magnitude of the horizontal component. Only the vertical component will change.
$\Rightarrow$ ${v_y} = u\sin \theta - gt$
$\Rightarrow$ ${v_x} = u\cos \theta $
On solving further, we get,
$\left( {u\cos \theta \hat i + u\sin \theta \hat j} \right).\left( {u\cos \theta \hat i + \left( {u\sin \theta - gt} \right)\hat j} \right) = 0$
$\Rightarrow$ ${u^2}{\cos ^2}\theta + {u^2}{\sin ^2}\theta - u\sin \theta gt = 0$
$\Rightarrow$ $t = \dfrac{u}{{g\sin \theta }}$
Thus, we need to select the correct option.

So, the correct option is D.

Note: The common mistake during the evaluation is doing the evaluation. It should be done carefully, as cases are different in case of ground-to-ground projection and projection from a height.
Also, acceleration due to gravity $\left( {\text{g}} \right)$ always acts vertically downwards and there is no acceleration in horizontal direction unless mentioned.
Take the value of $\left( {\text{g}} \right)$ to be $9.8m/{s^2}$ if not mentioned in the question. Generally, the value is considered to be $10m/{s^2}$ for the sake of calculation.
A projectile is an object upon which the only force is gravity. Gravity acts to influence the vertical motion of the projectile, thus causing a vertical acceleration. The horizontal motion of the projectile is the result of the tendency of any object in motion to remain in motion at constant velocity. Due to the absence of horizontal forces, a projectile remains in motion with a constant horizontal velocity. Horizontal forces are not required to keep a projectile moving horizontally. The only force acting upon a projectile is gravity.