Question

A ball is projected from a point with a speed $v$, at a certain angle with the horizontal. From the same point and at the same instant, a person starts running with a constant speed $0.5v$, to catch the ball. If the person catches the ball after some time, then the angle of projection of the ball is
Identify the correct option from the following:

• A. 60$^\circ$
• B. 30$^\circ$
• C. 45$^\circ$
• D. 53$^\circ$

Hint:

In projectile problems involving catching, equate the horizontal distance traveled by the person to the range of the projectile, and solve for the angle.

Answer 1

The Correct Option is A

Step 1: Set up the problem
Ball's initial velocity: $v$, angle $\theta$. Horizontal component: $v_x = v \cos \theta$. Person's speed: $0.5v$. The person catches the ball when they reach the landing point at the same time the ball lands. Step 2: Time of flight and range
Time of flight: $t = \frac{2 v \sin \theta}{g}$. Range: $R = (v \cos \theta) t = \frac{v^2 \sin 2\theta}{g}$. Person's distance: $d = (0.5v) t$. Set $d = R$: $0.5v \cdot \frac{2 v \sin \theta}{g} = \frac{v^2 \sin 2\theta}{g}$, $v \sin \theta = v \sin 2\theta \cos \theta$, $\sin \theta = 2 \sin \theta \cos^2 \theta$, $\cos^2 \theta = \frac{1}{2}$, $\cos \theta = \frac{1}{\sqrt{2}}$, $\theta = 60^\circ$. Step 3: Match with options
The angle 60$^\circ$ matches option (1).