A particle is projected with velocity v at an angle `theta` aith horizontal. The average angle velocity of the particle from the point of projection to impact equals

Options

A. `(gcostheta)/(thetav)`
B. `(g)/(vsintheta)`
C. `(g)/(vtheta)`
D. `(g theta)/(vsintheta)`

Answer

To find the average angular velocity of a particle projected with velocity \( v \) at an angle \( \theta \) with the horizontal, we can follow these steps:

### Step 1: Understand the motion of the particle
When a particle is projected at an angle \( \theta \), it follows a parabolic trajectory. The particle will strike the ground at the same angle \( \theta \) but in the opposite direction.

### Step 2: Determine the total angle rotated
The particle rotates from the initial angle \( \theta \) to the final angle \( -\theta \) upon impact. Therefore, the total angle rotated by the particle is:
\[ \Delta \theta = \theta - (-\theta) = 2\theta \]

### Step 3: Calculate the time of flight
The time of flight \( T \) for a projectile launched at an angle \( \theta \) with initial velocity \( v \) is given by the formula:
\[ T = \frac{2v \sin \theta}{g} \]
where \( g \) is the acceleration due to gravity.

### Step 4: Calculate the average angular velocity
The average angular velocity \( \omega_{avg} \) is defined as the total angle rotated divided by the total time taken:
\[ \omega_{avg} = \frac{\Delta \theta}{T} \]
Substituting the values we have:
\[ \omega_{avg} = \frac{2\theta}{\frac{2v \sin \theta}{g}} = \frac{2\theta \cdot g}{2v \sin \theta} \]
This simplifies to:
\[ \omega_{avg} = \frac{g \theta}{v \sin \theta} \]

### Conclusion
Thus, the average angular velocity of the particle from the point of projection to the point of impact is:
\[ \omega_{avg} = \frac{g \theta}{v \sin \theta} \]

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