What is the path followed by a moving body, on which a constant force acts in a direction other than initial velocity (i.e. excluding parallel and antiparallel direction) ?

Options

A. Straight line
B. Parabolic
C. Circular
D. Elliptical

Answer

To solve the problem of determining the path followed by a moving body under the influence of a constant force that acts in a direction other than the initial velocity, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- We have a moving body with an initial velocity \( \mathbf{u} \).
- A constant force \( \mathbf{F} \) acts on the body in a direction that is neither parallel nor anti-parallel to the initial velocity.

2. **Identifying the Forces**:
- The constant force can be represented as \( \mathbf{F} = m\mathbf{g} \) (where \( m \) is the mass of the body and \( \mathbf{g} \) is the acceleration due to gravity).
- The direction of the force is at an angle \( \theta \) with respect to the initial velocity \( \mathbf{u} \).

3. **Analyzing the Motion**:
- Since the force is acting at an angle, it will cause the body to accelerate in a direction that is different from its initial motion.
- The body will experience two components of motion: horizontal and vertical.

4. **Setting Up the Equations of Motion**:
- For horizontal motion (assuming no horizontal acceleration):
\[ x = u \cos(\theta) \cdot t \]
- For vertical motion (considering the downward force due to gravity):
\[ y = u \sin(\theta) \cdot t - \frac{1}{2} g t^2 \]

5. **Eliminating Time**:
- From the horizontal motion equation, we can express time \( t \) as:
\[ t = \frac{x}{u \cos(\theta)} \]
- Substitute this expression for \( t \) into the vertical motion equation:
\[ y = u \sin(\theta) \cdot \left(\frac{x}{u \cos(\theta)}\right) - \frac{1}{2} g \left(\frac{x}{u \cos(\theta)}\right)^2 \]

6. **Simplifying the Equation**:
- This leads to:
\[ y = x \tan(\theta) - \frac{g x^2}{2 u^2 \cos^2(\theta)} \]
- This equation represents a quadratic function in \( x \), indicating that the path followed by the body is a **parabola**.

7. **Conclusion**:
- Therefore, the path followed by the moving body under the influence of a constant force acting at an angle (neither parallel nor anti-parallel) to the initial velocity is a **parabolic trajectory**.