A particle is projected vertically upwards and it reaches the maximum height H in time T seconds. The height of the particle at any time t will be-

Options

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

Answer

To find the height of a particle projected vertically upwards at any time \( t \), we can use the equations of motion under uniform acceleration due to gravity. Here’s a step-by-step solution:

### Step 1: Understand the motion
When a particle is projected upwards, it will rise to a maximum height \( H \) in time \( T \) seconds. At this maximum height, its final velocity \( v = 0 \).

### Step 2: Relate initial velocity and time
Using the first equation of motion:
\[ v = u - gT \]
Since \( v = 0 \) at maximum height, we can set up the equation:
\[ 0 = u - gT \]
From this, we can express the initial velocity \( u \) as:
\[ u = gT \]

### Step 3: Write the equation for height at any time \( t \)
Using the second equation of motion, the height \( h \) at any time \( t \) is given by:
\[ h = ut - \frac{1}{2}gt^2 \]
Substituting \( u = gT \) into the equation:
\[ h = (gT)t - \frac{1}{2}gt^2 \]
This simplifies to:
\[ h = gtT - \frac{1}{2}gt^2 \]

### Step 4: Factor out \( g \)
Now, we can factor out \( g \):
\[ h = g\left(Tt - \frac{1}{2}t^2\right) \]

### Step 5: Rewrite the equation
To express \( h \) in terms of maximum height \( H \), we know that at \( t = T \):
\[ H = gT^2 - \frac{1}{2}gT^2 = \frac{1}{2}gT^2 \]
Thus, we can express the height \( h \) as:
\[ h = H - \frac{g}{2}(T - t)^2 \]

### Final Expression
The final expression for the height \( h \) of the particle at any time \( t \) is:
\[ h = H - \frac{g}{2}(T - t)^2 \]

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