Two balls of equal mass are projected from a tower simultaneously with equal speeds. One at angle `theta` above the horizontal and the other at the same angle `theta` below the horizontal. The path of the center of mass of the two balls is

Options

A. a vertical straight line
B. a horizontal straight line
C. a straight line at an angle `alpha (lt0)` with horizontal
D. a parabola

Explanation

To solve the problem of finding the path of the center of mass of two balls projected from a tower at angles θ above and below the horizontal, we can follow these steps:

### Step 1: Define the Masses and Velocities
Let the mass of each ball be \( m \). The initial velocity of both balls is \( v \). The first ball is projected at an angle \( \theta \) above the horizontal, and the second ball is projected at the same angle \( \theta \) below the horizontal.

### Step 2: Resolve the Velocities into Components
For the ball projected upwards:
- Horizontal component of velocity, \( v_{1x} = v \cos \theta \)
- Vertical component of velocity, \( v_{1y} = v \sin \theta \)

For the ball projected downwards:
- Horizontal component of velocity, \( v_{2x} = v \cos \theta \)
- Vertical component of velocity, \( v_{2y} = -v \sin \theta \) (negative because it is downward)

### Step 3: Calculate the Velocity of the Center of Mass
The velocity of the center of mass (CM) in the x-direction is given by:
\[ V_{CMx} = \frac{m v_{1x} + m v_{2x}}{m + m} = \frac{m(v \cos \theta) + m(v \cos \theta)}{2m} = v \cos \theta \]

The velocity of the center of mass in the y-direction is given by:
\[ V_{CMy} = \frac{m v_{1y} + m v_{2y}}{m + m} = \frac{m(v \sin \theta) + m(-v \sin \theta)}{2m} = 0 \]

### Step 4: Calculate the Acceleration of the Center of Mass
Since there are no horizontal forces acting on the balls, the acceleration in the x-direction is:
\[ A_{CMx} = 0 \]

In the y-direction, both balls experience gravitational acceleration \( g \) downwards, so:
\[ A_{CMy} = \frac{m(-g) + m(-g)}{m + m} = -g \]

### Step 5: Analyze the Motion of the Center of Mass
The center of mass moves with a constant velocity in the x-direction \( v \cos \theta \) and has a constant acceleration \( -g \) in the y-direction. This indicates that the motion of the center of mass is parabolic.

### Conclusion
The path of the center of mass of the two balls is a parabola.