From the concept of directed dimension what is the formula for a range (R) of a cannon ball when it is fired with vertical velocity component `V_(y)` and a horizontal velocity component `V_(x)`, assuming it is fired on a flat surface. [Range also depends upon acceleration due to gravity , g and k is numerical constant]

Options

A. `R=(k(V_(x)V_(y)))/(g)`
B. `R=(k(V_(x))^(2))/(g)`
C. `R=(k(V_(x))^(3))/(V_(y)g)`
D. `R=(k(V_(y))^(3))/(V_(x)g)`

Answer

To derive the formula for the range \( R \) of a cannonball fired with vertical velocity component \( V_y \) and horizontal velocity component \( V_x \), we will follow these steps:

### Step 1: Understand the Components of Velocity
The cannonball is fired with two components of velocity:
- Horizontal component: \( V_x \)
- Vertical component: \( V_y \)

### Step 2: Determine the Time of Flight
The time of flight \( t \) can be determined using the vertical motion of the cannonball. The vertical motion is influenced by gravity \( g \). The vertical velocity changes from \( V_y \) to \( -V_y \) when it reaches the maximum height and then falls back down.

Using the first equation of motion:
\[ V_{final} = V_{initial} + a \cdot t \]
Here, \( V_{final} = -V_y \), \( V_{initial} = V_y \), and \( a = -g \) (acceleration due to gravity).

Thus, we have:
\[ -V_y = V_y - g \cdot t \]
Rearranging gives:
\[ g \cdot t = 2V_y \implies t = \frac{2V_y}{g} \]

### Step 3: Calculate the Range
The range \( R \) is the horizontal distance traveled by the cannonball during the time of flight. The horizontal distance is given by:
\[ R = V_x \cdot t \]
Substituting the expression for \( t \):
\[ R = V_x \cdot \frac{2V_y}{g} \]

### Step 4: Introduce the Numerical Constant \( k \)
In the problem, we are told that the range also depends on a numerical constant \( k \). Therefore, we can express the range as:
\[ R = k \cdot \frac{2V_x V_y}{g} \]

### Step 5: Final Expression
Thus, we can write the final formula for the range \( R \) as:
\[ R = k \cdot \frac{V_x V_y}{g} \]

### Conclusion
The derived formula for the range \( R \) of a cannonball fired with vertical and horizontal components of velocity is:
\[ R = k \cdot \frac{V_x V_y}{g} \]