A projectile is projected with speed u at an angle `theta` with the horizontal . The average velocity of the projectile between the instants it crosses the same level is

Options

A. u cos `theta`
B. u sin `theta`
C. u cot `theta`
D. u tan `theta`

Answer

To find the average velocity of a projectile between the instants it crosses the same level, we can follow these steps:

### Step 1: Understand the Problem
A projectile is launched with an initial speed \( u \) at an angle \( \theta \) to the horizontal. We need to determine the average velocity as it returns to the same vertical level from which it was launched.

### Step 2: Define Average Velocity
The average velocity \( v_{avg} \) is defined as the total displacement divided by the total time taken. Mathematically, it can be expressed as:
\[ v_{avg} = \frac{\text{Total Displacement}}{\text{Total Time}} \]

### Step 3: Determine Total Displacement
Since the projectile returns to the same level from which it was launched, the total displacement in the vertical direction is zero. However, we are interested in the horizontal displacement (range) when it crosses the same level. The range \( R \) of the projectile can be calculated using the formula:
\[ R = \frac{u^2 \sin(2\theta)}{g} \]
where \( g \) is the acceleration due to gravity.

### Step 4: Calculate Total Time of Flight
The total time of flight \( T \) for the projectile can be calculated using the formula:
\[ T = \frac{2u \sin(\theta)}{g} \]

### Step 5: Substitute Values into Average Velocity Formula
Now, substituting the values of total displacement (which is the range \( R \)) and total time of flight \( T \) into the average velocity formula:
\[ v_{avg} = \frac{R}{T} = \frac{\frac{u^2 \sin(2\theta)}{g}}{\frac{2u \sin(\theta)}{g}} \]

### Step 6: Simplify the Expression
By simplifying the expression:
\[ v_{avg} = \frac{u^2 \sin(2\theta)}{g} \cdot \frac{g}{2u \sin(\theta)} = \frac{u \sin(2\theta)}{2 \sin(\theta)} \]
Using the identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \):
\[ v_{avg} = \frac{u \cdot 2 \sin(\theta) \cos(\theta)}{2 \sin(\theta)} = u \cos(\theta) \]

### Final Result
Thus, the average velocity of the projectile between the instants it crosses the same level is:
\[ v_{avg} = u \cos(\theta) \]