Choose the correct statement (s) :

Options

A. Whenpowerisconstant,thenvariationofforcewithvelocityishyperbolic.
B. Whenconstantpowerisappliedonthestationarybodyintimet,thenvariationofvelocitywithtimeisparabolic.
C. Whenpowerisconstant,thenvariationofvelocitywithtimeislinear.
D. Noneofthese.

Answer

To solve the question, we need to analyze each statement regarding the relationship between power, force, velocity, and time. Let's break down the statements one by one.

### Step-by-Step Solution:

1. **Understanding Power**:
Power (P) is defined as the rate at which work is done. It can also be expressed as the product of force (F) and velocity (v):
\[ P = F \cdot v \]
If power is constant, then the product of force and velocity remains constant.

2. **Analyzing Statement 1**:
"When power is constant, then variation of force with velocity is hyperbolic."
- Since \( P = F \cdot v \) is constant, if we plot force (F) against velocity (v), the relationship can be expressed as:
\[ F = \frac{P}{v} \]
This equation indicates that as velocity increases, force decreases, and vice versa, which is a hyperbolic relationship.
- **Conclusion**: This statement is **correct**.

3. **Analyzing Statement 2**:
"When constant power is applied on a stationary body, in time t, the variation of velocity with time is parabolic."
- If the body starts from rest (initial velocity \( u = 0 \)), and power is constant, then the force applied will change over time. As the body accelerates, the velocity increases, and the relationship between velocity and time can be derived from the equations of motion.
- The relationship \( v = u + at \) (where \( a \) is the acceleration) will yield a parabolic graph when plotted against time since the acceleration will change as the velocity increases.
- **Conclusion**: This statement is **correct**.

4. **Analyzing Statement 3**:
"When power is constant, then variation of velocity with time is linear."
- If power is constant, then the force applied is not constant (it varies with velocity). Therefore, the acceleration (which is force divided by mass) will also not be constant. This means that the velocity will not change linearly with time.
- **Conclusion**: This statement is **incorrect**.

5. **Analyzing Statement 4**:
"None of these."
- Since we have established that the first two statements are correct, this statement must be **incorrect**.

### Final Conclusion:
The correct statements are:
- Statement 1: Correct
- Statement 2: Correct
- Statement 3: Incorrect
- Statement 4: Incorrect

Thus, the correct answers are **Statement 1 and Statement 2**.