The ratio of momentum and kinetic energy of particle is inversely proportional to the time. Then, this is the case of a

Options

A. uniformly accelerated motion
B. uniform motion
C. uniformly retarted motion
D. simple harmonic motion

Answer

To solve the problem, we need to analyze the relationship between momentum, kinetic energy, and time as described in the question. Let's break it down step by step.

### Step 1: Understand the Definitions
- **Momentum (p)** of a particle is given by the formula:
\[ p = mv \]
where \( m \) is the mass and \( v \) is the velocity of the particle.

- **Kinetic Energy (KE)** of a particle is given by the formula:
\[ KE = \frac{1}{2} mv^2 \]

### Step 2: Set Up the Ratio
According to the question, the ratio of momentum to kinetic energy is inversely proportional to time \( t \). We can express this relationship mathematically:
\[ \frac{p}{KE} \propto \frac{1}{t} \]

### Step 3: Substitute the Definitions
Now, substitute the expressions for momentum and kinetic energy into the ratio:
\[ \frac{p}{KE} = \frac{mv}{\frac{1}{2} mv^2} \]

### Step 4: Simplify the Ratio
Cancel out the mass \( m \) from the numerator and denominator:
\[ \frac{p}{KE} = \frac{mv}{\frac{1}{2} mv^2} = \frac{2}{v} \]

### Step 5: Establish the Proportionality
From the above, we have:
\[ \frac{2}{v} \propto \frac{1}{t} \]
This implies:
\[ v \propto t \]
This means that velocity \( v \) is directly proportional to time \( t \).

### Step 6: Analyze the Motion
If velocity is directly proportional to time, it indicates that the object is undergoing uniformly accelerated motion. In uniformly accelerated motion, the velocity increases linearly with time.

### Conclusion
Thus, the case described in the question is that of **uniformly accelerated motion**.