A particle is acted upon by a constant power. Then, which of the following physical quantity remains constant

Options

A. Speed
B. Rate of change of acceleration
C. Kinetic energy
D. Rate of change of kinetic energy

Explanation

To solve the problem, we need to analyze the relationship between power, work, and kinetic energy when a constant power is applied to a particle.

### Step-by-Step Solution:

1. **Understanding Power**:
Power (P) is defined as the rate at which work is done. Mathematically, it can be expressed as:
\[ P = \frac{dW}{dt} \]
where \(dW\) is the infinitesimal work done, and \(dt\) is the infinitesimal time interval.

2. **Constant Power Implication**:
If the power is constant, it means that the rate of doing work is constant. Therefore, we can say:
\[ P = \text{constant} \]

3. **Relating Power to Kinetic Energy**:
According to the work-energy theorem, the work done on a particle is equal to the change in its kinetic energy (KE):
\[ W = \Delta KE \]
Thus, we can express the change in kinetic energy in terms of power:
\[ P = \frac{d(\Delta KE)}{dt} \]
Since \(P\) is constant, it follows that:
\[ \frac{d(\Delta KE)}{dt} = \text{constant} \]

4. **Conclusion about Kinetic Energy**:
The above equation indicates that the rate of change of kinetic energy is constant. This does not mean that the kinetic energy itself is constant; rather, it means that the change in kinetic energy over time is constant.

5. **Evaluating the Options**:
- **Option 1**: Speed - Not necessarily constant.
- **Option 2**: Rate of change of acceleration - Not relevant here.
- **Option 3**: Kinetic energy - Not constant, but changing at a constant rate.
- **Option 4**: Rate of change of kinetic energy - This is constant.

Thus, the correct answer is that the **rate of change of kinetic energy remains constant** when a constant power acts on the particle.

### Final Answer:
**The physical quantity that remains constant is the rate of change of kinetic energy.**

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