Question

The speed of a projectile at its maximum height is half of its initial speed. The angle of projection is-
A. 60 degrees
B. 15 degrees
C. 45 degrees
D. 30 degrees

Answer 1

Hint: We will acknowledge the fact that the value of the vertical velocity is zero when the object is at its highest point. We will name the two components i.e. the vertical and the horizontal component and then solve the solution further. Refer to the solution below.

Step-By-Step answer:
Let the velocity of the ball projected at the $\theta$ be u. The ball will go through projectile motion (see the figure).
As it is already said in the question that the speed of the ball will be half its initial speed once it reaches its maximum height. Let the speed at the point when the ball is at its maximum height be v.
When the above statement which says- the velocity when the ball is at its maximum height is half the initial speed- will be written mathematically, we will have-
$\Rightarrow v = \dfrac{u}{2}$
Let this be equation 1-
$\Rightarrow v = \dfrac{u}{2}$ (equation 1)
As we already know that the velocity in projectile motion has two components i.e. $u\cos \theta$ (horizontal component) and $u\sin \theta$ (vertical component).
As we can see that there is no force acting in the horizontal direction thus there will be no acceleration. Hence, the horizontal velocity will be the same at the initial point and the highest point. At the highest point, only horizontal velocity works and the vertical velocity becomes zero.
Thus, we can say that-
$\Rightarrow v = u\cos \theta$
Let this be equation 2-
$\Rightarrow v = u\cos \theta$ (equation 2)
Equating the equation 1 and equation 2, we get-
$\Rightarrow \dfrac{u}{2} = u\cos \theta \\\ \\\ \Rightarrow \dfrac{1}{2} = \cos \theta \\\ \\\ \Rightarrow \cos \theta = \dfrac{1}{2} \\\$
As we know that the value of $\cos \theta$ is 1/2 when the value of $\theta$ is 60 degrees.
Hence, option A is the correct option.

Note: Projectile motion is an object's or particle's type of motion (a projectile) that is projected on the surface of the earth and travels through a curved path only under the control of gravity (in fact, it is presumed that the air resistance effects are negligible). Galileo revealed that this twisted path is a parabola which can also be a line in the same case when tossed up. The analysis of such motions is called ballistics, and a ballistic pathway is like that.