Read the following statements and choose the correct answer.
(a) For a freely falling body the average velocity is proportional to square root of height of fall.
(b) For a freely falling body the displacements in successive equal time intervals are in the ratio 1:4:9:....
(c) For a vertically projected body the displacement during last second of time of flight changes with velocity of projection.
(d) For a body projected from the top of the tower the displacement of the body is negative when the body crosses the point of projection

Options

A. a,c,d are true
B. Both a,b are true
C. b and c are true
D. c and d are true

Answer

To solve the question, we need to analyze each of the statements provided regarding the motion of freely falling bodies and vertically projected bodies. Let's evaluate each statement one by one.

### Step 1: Analyze Statement (a)
**Statement (a)**: For a freely falling body, the average velocity is proportional to the square root of the height of fall.

- When a body is dropped from a height \( h \), the average velocity \( v_{avg} \) can be calculated using the formula:
\[ v_{avg} = \frac{h}{t} \]
where \( t \) is the time taken to fall. Using the second equation of motion,
\[ h = ut + \frac{1}{2}gt^2 \]
where \( u = 0 \) (initial velocity), we can find \( t \):
\[ h = \frac{1}{2}gt^2 \implies t = \sqrt{\frac{2h}{g}} \]
Substituting this back into the average velocity equation gives:
\[ v_{avg} = \frac{h}{\sqrt{\frac{2h}{g}}} = \sqrt{\frac{gh}{2}} \]
Thus, the average velocity is indeed proportional to the square root of the height \( h \). **This statement is true.**

### Step 2: Analyze Statement (b)
**Statement (b)**: For a freely falling body, the displacements in successive equal time intervals are in the ratio 1:4:9:...

- The displacement in successive time intervals can be calculated using:
\[ s_n = ut + \frac{1}{2}gt^2 \]
For \( t = 1, 2, 3 \):
- \( s_1 = \frac{1}{2}g(1^2) = \frac{g}{2} \)
- \( s_2 = \frac{1}{2}g(2^2) = 2g \)
- \( s_3 = \frac{1}{2}g(3^2) = \frac{9g}{2} \)

The displacements in successive intervals are:
- First interval: \( s_1 = \frac{g}{2} \)
- Second interval: \( s_2 - s_1 = 2g - \frac{g}{2} = \frac{3g}{2} \)
- Third interval: \( s_3 - s_2 = \frac{9g}{2} - 2g = \frac{5g}{2} \)

The ratios of displacements are not 1:4:9. **This statement is false.**

### Step 3: Analyze Statement (c)
**Statement (c)**: For a vertically projected body, the displacement during the last second of time of flight changes with the velocity of projection.

- The displacement during the last second can be calculated using the formula:
\[ s = u - \frac{1}{2}g \]
This clearly shows that the displacement depends on the initial velocity \( u \). **This statement is true.**

### Step 4: Analyze Statement (d)
**Statement (d)**: For a body projected from the top of the tower, the displacement of the body is negative when the body crosses the point of projection.

- If we take upward as positive, then when the body falls below the point of projection (the height from which it was thrown), the displacement will be negative. **This statement is true.**

### Conclusion
The correct statements are (a), (c), and (d). The only incorrect statement is (b).

### Final Answer
The correct answer is: **(a), (c), and (d) are true; (b) is false.**

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