A body is thrown vertically upward with velocity `u`, the greatest height `h` to which it will rise is,

Options

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

Explanation

To find the greatest height \( h \) to which a body will rise when thrown vertically upward with an initial velocity \( u \), we can use the equations of motion. Here’s a step-by-step solution:

### Step 1: Understand the motion
When a body is thrown vertically upward, it will rise until it reaches its maximum height, at which point its velocity will become zero.

### Step 2: Identify the known values
- Initial velocity \( u \) (the velocity with which the body is thrown upward)
- Final velocity \( v = 0 \) m/s (at the maximum height)
- Acceleration \( a = -g \) (the acceleration due to gravity, acting downwards)

### Step 3: Use the third equation of motion
The third equation of motion relates the initial velocity, final velocity, acceleration, and displacement:
\[ v^2 = u^2 + 2as \]
In this case:
- \( v = 0 \)
- \( u = u \)
- \( a = -g \)
- \( s = h \) (the height we want to find)

Substituting these values into the equation gives:
\[ 0 = u^2 + 2(-g)h \]

### Step 4: Rearranging the equation
Rearranging the equation to solve for \( h \):
\[ 0 = u^2 - 2gh \]
\[ 2gh = u^2 \]
\[ h = \frac{u^2}{2g} \]

### Step 5: Conclusion
The greatest height \( h \) to which the body will rise is given by:
\[ h = \frac{u^2}{2g} \]