A particle is moving in a circular path. The acceleration and momentum vector at an instant of time are `veca= 2hat i+ 3 hat j m//sec^(2)` and `vec p= 6 hat i-4 hatj kg m//sec`. Then the motion of the particle is

Options

A. Uniform circular motion
B. Circular motion with tangential acceleration
C. Circular motion with tangential retardation
D. We can not say anything from `veca` and `vecp` only

Answer

To determine the nature of the motion of the particle, we need to analyze the given vectors for acceleration and momentum. Let's break down the solution step by step.

### Step 1: Identify the vectors
We are given:
- Acceleration vector: \(\vec{a} = 2 \hat{i} + 3 \hat{j} \, \text{m/s}^2\)
- Momentum vector: \(\vec{p} = 6 \hat{i} - 4 \hat{j} \, \text{kg m/s}\)

### Step 2: Calculate the magnitudes of the vectors
The magnitude of the acceleration vector \(\vec{a}\) is calculated as follows:
\[ |\vec{a}| = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \, \text{m/s}^2 \]

The magnitude of the momentum vector \(\vec{p}\) is calculated as follows:
\[ |\vec{p}| = \sqrt{(6)^2 + (-4)^2} = \sqrt{36 + 16} = \sqrt{52} \, \text{kg m/s} \]

### Step 3: Calculate the dot product of the vectors
The dot product \(\vec{a} \cdot \vec{p}\) is calculated as follows:
\[ \vec{a} \cdot \vec{p} = (2)(6) + (3)(-4) = 12 - 12 = 0 \]

### Step 4: Determine the angle between the vectors
The dot product is related to the angle \(\theta\) between the two vectors by the formula:
\[ \vec{a} \cdot \vec{p} = |\vec{a}| |\vec{p}| \cos \theta \]
Since we found that \(\vec{a} \cdot \vec{p} = 0\), we can conclude:
\[ |\vec{a}| |\vec{p}| \cos \theta = 0 \]
Given that neither \(|\vec{a}|\) nor \(|\vec{p}|\) is zero, it follows that:
\[ \cos \theta = 0 \implies \theta = 90^\circ \]

### Step 5: Interpret the result
An angle of \(90^\circ\) between the acceleration vector and the momentum vector indicates that the motion of the particle is purely circular. In this case, the acceleration is centripetal, and there is no tangential acceleration.

### Conclusion
The motion of the particle is uniform circular motion.

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