Which of the following statements is/are incorrect?
(i) Work and energy have different units.
(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 ofa particle is doubled, the ratio of its momentum and kinetic energy gets halved.

Options

A. (ii) only
B. (i) and (iii) only
C. (i), (ii) and (iv) only
D. (i), (ii), (iii) and (iv)

Answer

To determine which statements are incorrect, let's analyze each one step by step.

1. **Statement (i)**: "Work and energy have different units."
- **Analysis**: Work is defined as the product of force and displacement, and its unit is Joules (J). Energy, such as kinetic energy or potential energy, is also measured in Joules (J). Therefore, both work and energy share the same unit.
- **Conclusion**: This statement is **incorrect**.

2. **Statement (ii)**: "Two bodies of unequal masses have equal acceleration at any instant of time when they are dropped from a cliff."
- **Analysis**: According to the law of universal gravitation and Newton's second law of motion, all objects in free fall near the Earth's surface experience the same acceleration due to gravity (approximately 9.8 m/s²), regardless of their masses. Thus, two bodies dropped from a cliff will indeed have equal acceleration.
- **Conclusion**: This statement is **correct**.

3. **Statement (iii)**: "When an aeroplane takes off, the work done by its weight is positive."
- **Analysis**: When an airplane takes off, it moves upwards against the force of gravity. The weight of the airplane acts downwards while the displacement during takeoff is upwards. The work done by a force is given by the formula: Work = Force × Displacement × cos(θ), where θ is the angle between the force and displacement vectors. Since the angle is 180° (force downwards and displacement upwards), the work done by weight is negative.
- **Conclusion**: This statement is **incorrect**.

4. **Statement (iv)**: "When the speed of a particle is doubled, the ratio of its momentum and kinetic energy gets halved."
- **Analysis**: Momentum (p) is given by the formula p = mv, where m is mass and v is velocity. Kinetic energy (KE) is given by KE = 1/2 mv². If the speed is doubled (v becomes 2v), then:
- New momentum = m(2v) = 2mv
- New kinetic energy = 1/2 m(2v)² = 1/2 m(4v²) = 2mv²
- The ratio of momentum to kinetic energy is:
\[ \text{Ratio} = \frac{p}{KE} = \frac{mv}{\frac{1}{2}mv^2} = \frac{2}{v} \]
- When speed is doubled, the new ratio becomes:
\[ \text{New Ratio} = \frac{2mv}{2mv^2} = \frac{2}{2v} = \frac{1}{v} \]
- Hence, the ratio does not get halved; it changes in a different manner.
- **Conclusion**: This statement is **incorrect**.

### Summary of Incorrect Statements:
- Statement (i) is incorrect.
- Statement (iii) is incorrect.
- Statement (iv) is incorrect.