A particle is projected vertically upwards and reaches the maximum height H in time T. The height of the partlcle at any time `t (lt T)` will be

Options

A. `g (t-T)^2`
B. `H - g (t-T)^2`
C. `1/2 g (t-T)^2`
D. `H - 1/2 g (T-t)^2`

Explanation

To solve the problem of finding the height of a particle projected vertically upwards at any time \( t \) (where \( t < T \)), we can follow these steps:

### Step 1: Understand the motion of the particle
When a particle is projected upwards, it moves against the force of gravity until it reaches its maximum height \( H \) at time \( T \). At this point, its velocity becomes zero.

### Step 2: Use the equations of motion
We can use the second equation of motion to find the distance covered by the particle in time \( t \):
\[ s = ut + \frac{1}{2} a t^2 \]
where:
- \( s \) is the distance traveled,
- \( u \) is the initial velocity,
- \( a \) is the acceleration (which is \( -g \) for upward motion),
- \( t \) is the time.

Since the particle is projected upwards, the initial velocity \( u \) can be expressed in terms of the maximum height \( H \) and time \( T \):
\[ u = gT \]
This is because the particle will have a velocity \( u \) at \( t = 0 \) and will decelerate due to gravity until it reaches the maximum height.

### Step 3: Calculate the distance traveled in time \( t \)
Using the equation of motion:
\[ s = gTt - \frac{1}{2} g t^2 \]
This gives us the distance traveled upwards in time \( t \).

### Step 4: Find the height at time \( t \)
The height \( h \) of the particle at time \( t \) can be calculated as:
\[ h = H - s \]
Substituting for \( s \):
\[ h = H - \left( gTt - \frac{1}{2} g t^2 \right) \]
Since \( H = \frac{1}{2} g T^2 \) (the total height reached at time \( T \)), we can substitute \( H \) into the equation:
\[ h = \frac{1}{2} g T^2 - \left( gTt - \frac{1}{2} g t^2 \right) \]
Simplifying this:
\[ h = \frac{1}{2} g T^2 - gTt + \frac{1}{2} g t^2 \]
\[ h = \frac{1}{2} g (T^2 - 2Tt + t^2) \]
\[ h = \frac{1}{2} g (T - t)^2 \]

### Final Equation
Thus, the height of the particle at any time \( t \) is given by:
\[ h = H - \frac{g}{2} (T - t)^2 \]