Which of the following statements is/are incorrect?
I. Area under the graph between displacement and force applied gives the work done.
II. Two bodies of unequal masses have equal acceleration at any instant of time when they are dropped from a cliff.
III. When an aeroplane takes off, the work done by its weight is positive.
IV. When the speed of a particle is halved, the ratio of its momentum and kinetic energy also gets halved.

Options

A. II only
B. III and IV only
C. I, II and IV only
D. I, II, III and IV

Answer

To determine which statements are incorrect, let's analyze each statement one by one:

### Step 1: Analyze Statement I
**Statement I:** Area under the graph between displacement and force applied gives the work done.

- The work done (W) is defined as the integral of force (F) with respect to displacement (s), which is mathematically expressed as:
\[ W = \int F \, ds \]
- The area under the force vs. displacement graph represents this integral. Therefore, this statement is **correct**.

### Step 2: Analyze Statement II
**Statement II:** Two bodies of unequal masses have equal acceleration at any instant of time when they are dropped from a cliff.

- According to Newton's second law, the acceleration of an object in free fall is independent of its mass. Both bodies will experience the same gravitational acceleration (approximately \(9.8 \, m/s^2\) near the Earth's surface).
- Hence, this statement is **correct**.

### Step 3: Analyze Statement III
**Statement III:** When an aeroplane takes off, the work done by its weight is positive.

- When an airplane takes off, it moves upward against the force of gravity. The weight of the airplane acts downward, while the displacement during takeoff is upward.
- Work done (W) is defined as:
\[ W = F \cdot d \cdot \cos(\theta) \]
where \(\theta\) is the angle between the force and the displacement. In this case, \(\theta = 180^\circ\) (since weight acts downward and displacement is upward), thus:
\[ W = F \cdot d \cdot \cos(180^\circ) = -F \cdot d \]
- Therefore, the work done by the weight of the airplane is negative. This statement is **incorrect**.

### Step 4: Analyze Statement IV
**Statement IV:** When the speed of a particle is halved, the ratio of its momentum and kinetic energy also gets halved.

- The momentum (p) of an object is given by:
\[ p = mv \]
- The kinetic energy (KE) is given by:
\[ KE = \frac{1}{2} mv^2 \]
- If the speed is halved (v becomes \(v/2\)):
- New momentum:
\[ p' = m \left(\frac{v}{2}\right) = \frac{mv}{2} \]
- New kinetic energy:
\[ KE' = \frac{1}{2} m \left(\frac{v}{2}\right)^2 = \frac{1}{2} m \left(\frac{v^2}{4}\right) = \frac{mv^2}{8} \]
- The ratio of momentum to kinetic energy is:
\[ \text{Ratio} = \frac{p'}{KE'} = \frac{\frac{mv}{2}}{\frac{mv^2}{8}} = \frac{8}{2v} = \frac{4}{v} \]
- This shows that the ratio does not simply halve; hence, this statement is also **incorrect**.

### Conclusion
The incorrect statements are:
- Statement III: When an aeroplane takes off, the work done by its weight is positive.
- Statement IV: When the speed of a particle is halved, the ratio of its momentum and kinetic energy also gets halved.

### Final Answer
**Incorrect Statements:** III and IV.

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