Height of the body from the ground can be calculated by using the formula `h=-ut+(1//2)(g t^2)` in
(a) A body projected vertically with velocity 'u' from the top of tower, reaches the ground in 't' sec.
( b) A body dropped from a balloon moving up with uniform velocity, reaches the ground in 't' sec
(c) A body dropped from a helicopter moving up with uniform velocity, reaches the ground in 't' sec
(d) A body projected vertically from the ground reaches the ground in 't' sec.

Options

A. a, b and c are correct
B. a, b, c and d are correct
C. a is only correct
D. b and d are correct

Answer

To solve the problem, we will analyze each option using the formula for height \( h \) given by:

\[ h = -ut + \frac{1}{2}gt^2 \]

where:
- \( h \) = height from the ground,
- \( u \) = initial velocity,
- \( g \) = acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
- \( t \) = time in seconds.

### Step-by-step Solution:

#### (a) A body projected vertically with velocity 'u' from the top of a tower, reaches the ground in 't' sec.

1. Here, the body is projected upward with an initial velocity \( u \).
2. The height \( h \) can be calculated using the formula:
\[ h = -ut + \frac{1}{2}gt^2 \]
3. Since the body is projected upwards, we consider \( u \) as positive, and the height will be positive.
4. This option is valid as it correctly describes the motion.

**Conclusion**: This option is valid.

#### (b) A body dropped from a balloon moving up with uniform velocity, reaches the ground in 't' sec.

1. The balloon is moving upward with a uniform velocity \( u \).
2. When the body is dropped, its initial velocity will be \( -u \) (since it is moving in the opposite direction).
3. The height can be calculated as:
\[ h = -(-u)t + \frac{1}{2}gt^2 = ut + \frac{1}{2}gt^2 \]
4. This option is valid as it correctly describes the motion.

**Conclusion**: This option is valid.

#### (c) A body dropped from a helicopter moving up with uniform velocity, reaches the ground in 't' sec.

1. Similar to option (b), the helicopter is moving upward with a uniform velocity \( u \).
2. When the body is dropped, its initial velocity will also be \( -u \).
3. The height can be calculated as:
\[ h = -(-u)t + \frac{1}{2}gt^2 = ut + \frac{1}{2}gt^2 \]
4. This option is valid as it correctly describes the motion.

**Conclusion**: This option is valid.

#### (d) A body projected vertically from the ground reaches the ground in 't' sec.

1. Here, the body is projected upward from the ground with an initial velocity \( u \).
2. The height can be calculated using the formula:
\[ h = -ut + \frac{1}{2}gt^2 \]
3. This option is valid as it correctly describes the motion.

**Conclusion**: This option is valid.

### Final Conclusion:
All options (a), (b), (c), and (d) are valid. The correct answer is that options (b) and (d) are particularly highlighted as correct in the context of the problem.