A shell is fired vertically upwards with a velocity `v_(1) ` from a trolley moving horizontally with velocity `v_(2)`. A person on the the ground observes the motion of the shell as a parabole, whose horizontal range is

Options

A. `(2v_(1)^(2)v_(2))/(g)`
B. `(2v_(1)^(2))/(g)`
C. `(2v_(2)^(2))/(g)`
D. `(2v_(1)v_(2))/(g)`

Answer

To solve the problem of finding the horizontal range of a shell fired vertically upwards from a trolley moving horizontally, we can break down the solution into the following steps:

### Step 1: Understand the Motion
The shell is fired vertically upwards with a velocity \( v_1 \) from a trolley that is moving horizontally with a velocity \( v_2 \). The motion of the shell can be analyzed in two dimensions: vertical and horizontal.

**Hint:** Remember that the horizontal and vertical motions are independent of each other.

### Step 2: Identify the Components of Velocity
- The horizontal component of the shell's velocity (\( u_x \)) is equal to the velocity of the trolley, which is \( v_2 \).
- The vertical component of the shell's velocity (\( u_y \)) is equal to the velocity with which it is fired upwards, which is \( v_1 \).

**Hint:** Write down the equations for horizontal and vertical components of motion separately.

### Step 3: Determine the Time of Flight
The time of flight (\( t \)) for the shell can be determined by analyzing the vertical motion. The time taken for the shell to reach its maximum height and return to the original height can be calculated using the formula:
\[ t = \frac{2u_y}{g} = \frac{2v_1}{g} \]
where \( g \) is the acceleration due to gravity.

**Hint:** Use the kinematic equations for vertical motion to find the time of flight.

### Step 4: Calculate the Horizontal Range
The horizontal range (\( R \)) can be calculated using the horizontal velocity and the time of flight:
\[ R = u_x \cdot t = v_2 \cdot t \]
Substituting the expression for \( t \):
\[ R = v_2 \cdot \frac{2v_1}{g} \]
Thus, the horizontal range is:
\[ R = \frac{2v_1 v_2}{g} \]

**Hint:** Remember that the horizontal range is the product of horizontal velocity and time of flight.

### Final Answer
The horizontal range of the shell observed by a person on the ground is:
\[ R = \frac{2v_1 v_2}{g} \]