A particle moves in a plane with a constant acceleration in a direction different from the initial velocity. The path of the particle is

Options

A. straight line
B. arc of circle
C. parabola
D. ellipse

Explanation

To solve the problem of determining the path of a particle moving in a plane with constant acceleration in a direction different from its initial velocity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let the initial velocity of the particle be \( \vec{v} \).
- Let the constant acceleration be \( \vec{a} \).
- Let the angle between the initial velocity and acceleration be \( \theta \).

2. **Break Down the Velocity and Acceleration**:
- The initial velocity can be broken down into two components:
- \( v_x = v \cos \theta \) (horizontal component)
- \( v_y = v \sin \theta \) (vertical component)

3. **Apply Kinematic Equations**:
- The vertical motion can be described by the equation:
\[ y = v_y t + \frac{1}{2} a_y t^2 \]
- Here, \( a_y \) is the vertical component of acceleration. Since the acceleration is constant and directed differently, we can assume \( a_y = a \).

4. **Substituting the Components**:
- Substitute \( v_y \) and \( a_y \) into the equation:
\[ y = (v \sin \theta) t + \frac{1}{2} a t^2 \]

5. **Horizontal Motion**:
- The horizontal motion is given by:
\[ x = v_x t \]
- Substitute \( v_x \):
\[ x = (v \cos \theta) t \]
- From this, we can express time \( t \) in terms of \( x \):
\[ t = \frac{x}{v \cos \theta} \]

6. **Substituting Time into the Vertical Equation**:
- Substitute \( t \) back into the vertical motion equation:
\[ y = (v \sin \theta) \left(\frac{x}{v \cos \theta}\right) + \frac{1}{2} a \left(\frac{x}{v \cos \theta}\right)^2 \]
- Simplifying this gives:
\[ y = x \tan \theta + \frac{1}{2} a \frac{x^2}{v^2 \cos^2 \theta} \]

7. **Identifying the Path**:
- The equation derived is of the form:
\[ y = Ax + Bx^2 \]
- This is a quadratic equation in \( x \), which represents a **parabola**.

### Conclusion:
The path of the particle is a **parabola**.