Two equal masses (m) are projected at the same angle (θ) from two points separated by their range with equal velocities (v). The momentum at the point of their collision is

Correct Option: The correct answer is Zero.

Let us analyze the problem step-by-step:

1. Understanding the physical setup:
Two equal masses, each of mass $m$, are projected with equal initial speeds $v$ at the same angle $\theta$ relative to the horizontal. They are projected towards each other from two points on the ground separated by a distance equal to their horizontal range $R$.

2. Locating the collision point:
Since the particles are projected simultaneously with the same velocity and angle from two points separated by their range, by symmetry, they will meet and collide at the highest point of their trajectories. This collision point is located at a horizontal distance of $\frac{R}{2}$ from both projection points.

3. Determining velocities at the point of collision:
At the highest point of a projectile's path, the vertical component of velocity becomes zero.
The horizontal component of velocity remains constant throughout the flight because there is no horizontal force acting on the masses.

Let the horizontal direction from the first point to the second point be the positive x-direction, represented by the unit vector $\overrightarrow{i}$.

For the first mass (moving to the right):
${\overrightarrow{v}}_1=vcos\theta \overrightarrow{i}$

For the second mass (moving to the left):
${\overrightarrow{v}}_2=-vcos\theta \overrightarrow{i}$

4. Calculating the total momentum at the point of collision:
The total linear momentum $\overrightarrow{P}$ of the system at the instant of collision is the vector sum of the individual momenta of the two masses:

$\overrightarrow{P}=m{\overrightarrow{v}}_1+m{\overrightarrow{v}}_2$

Substituting the velocity vectors:

$\overrightarrow{P}=m(vcos\theta \overrightarrow{i})+m(-vcos\theta \overrightarrow{i})$

$\overrightarrow{P}=mvcos\theta \overrightarrow{i}-mvcos\theta \overrightarrow{i}=0$

Therefore, the total momentum of the system at the point of collision is zero.