A particular has initial velocity , `v=3hati+4hatj` and a constant force `F=4hati-3hatj` acts on it. The path of the particle is

Options

A. straight line
B. parabolic
C. circular
D. elliptical

Explanation

To determine the path of the particle given its initial velocity and the constant force acting on it, we can follow these steps:

### Step 1: Identify the initial velocity and force
The initial velocity of the particle is given as:
\[ \mathbf{v} = 3\hat{i} + 4\hat{j} \]
The constant force acting on the particle is:
\[ \mathbf{F} = 4\hat{i} - 3\hat{j} \]

### Step 2: Calculate the dot product of force and velocity
The dot product \(\mathbf{F} \cdot \mathbf{v}\) can be calculated using the formula:
\[ \mathbf{F} \cdot \mathbf{v} = F_x \cdot v_x + F_y \cdot v_y \]
Substituting the values:
\[ \mathbf{F} \cdot \mathbf{v} = (4)(3) + (-3)(4) = 12 - 12 = 0 \]

### Step 3: Analyze the result of the dot product
Since the dot product \(\mathbf{F} \cdot \mathbf{v} = 0\), this indicates that the force and the velocity are perpendicular to each other.

### Step 4: Determine the implications of perpendicular force and velocity
When the force is perpendicular to the velocity, it implies that the particle is undergoing a change in direction but not in speed. This can lead to two possible types of motion: circular or parabolic.

### Step 5: Determine the nature of the path
Since the force acting on the particle is constant (not varying in direction), the particle cannot follow a circular path. A circular path requires a centripetal force that changes direction continuously. Therefore, the only possibility left is that the particle follows a parabolic path.

### Conclusion
Thus, the path of the particle is a **parabolic path**.

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