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 find the height of the particle at any time \( t \) (where \( t < T \)), we can use the equations of motion.

### Step-by-Step Solution:

1. **Understanding the Motion**:
The particle is projected upwards with an initial velocity \( u \) and moves under the influence of gravity. At the maximum height \( H \), the velocity becomes zero.

2. **Time to Reach Maximum Height**:
The time \( T \) to reach the maximum height is given. At maximum height, the final velocity \( v = 0 \). Using the equation of motion:
\[ v = u - gT \]
Setting \( v = 0 \):
\[ 0 = u - gT \implies u = gT \]

3. **Maximum Height**:
The maximum height \( H \) can be calculated using the equation:
\[ H = uT - \frac{1}{2}gT^2 \]
Substituting \( u = gT \):
\[ H = (gT)T - \frac{1}{2}gT^2 = gT^2 - \frac{1}{2}gT^2 = \frac{1}{2}gT^2 \]

4. **Height at Time \( t \)**:
To find the height \( h(t) \) at any time \( t < T \), we use the same equation of motion:
\[ h(t) = ut - \frac{1}{2}gt^2 \]
Substituting \( u = gT \):
\[ h(t) = (gT)t - \frac{1}{2}gt^2 \]
Simplifying this:
\[ h(t) = gtT - \frac{1}{2}gt^2 = gTt - \frac{1}{2}gt^2 \]

5. **Final Expression**:
Thus, the height of the particle at any time \( t \) (where \( t < T \)) is given by:
\[ h(t) = gTt - \frac{1}{2}gt^2 \]