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 question regarding the path followed by a moving body when a constant force acts in a direction other than the initial velocity, we can break down the problem step by step.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to determine the path of a body that is moving under the influence of a constant force that is not aligned with its initial velocity. This means the force is acting at an angle to the direction of motion.

**Hint**: Visualize the scenario by drawing a diagram showing the initial velocity vector and the force vector at an angle.

2. **Applying Newton's Second Law**:
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). The acceleration will be in the direction of the net force acting on the body.

**Hint**: Remember that the direction of acceleration is determined by the direction of the net force.

3. **Considering Projectile Motion**:
Let's analyze the situation as a projectile motion where the body is projected at an angle θ with an initial velocity \( u \). The force acting on the body (like gravity) will influence its vertical motion.

**Hint**: Think of the body’s motion as being composed of horizontal and vertical components.

4. **Breaking Down the Motion**:
- The horizontal component of the motion remains constant since there is no horizontal force acting on it.
- The vertical component of the motion is influenced by the gravitational force acting downward.

**Hint**: Use \( u_x = u \cos \theta \) for horizontal and \( u_y = u \sin \theta \) for vertical components.

5. **Setting Up the Equations of Motion**:
- For horizontal motion:
\[ x = u \cos \theta \cdot t \quad \text{(Equation 1)} \]
- For vertical motion, using the second equation of motion:
\[ y = u \sin \theta \cdot t - \frac{1}{2} g t^2 \quad \text{(Equation 2)} \]

**Hint**: Remember to keep track of the time variable \( t \) in both equations.

6. **Eliminating Time**:
From Equation 1, we can express time \( t \) in terms of \( x \):
\[ t = \frac{x}{u \cos \theta} \]
Substitute this expression for \( t \) into Equation 2.

**Hint**: This substitution will help you relate \( y \) directly to \( x \).

7. **Substituting and Simplifying**:
Substitute \( t \) into Equation 2:
\[ y = u \sin \theta \left(\frac{x}{u \cos \theta}\right) - \frac{1}{2} g \left(\frac{x}{u \cos \theta}\right)^2 \]
Simplifying this gives:
\[ y = x \tan \theta - \frac{g x^2}{2 u^2 \cos^2 \theta} \]

**Hint**: This equation represents a quadratic relationship between \( y \) and \( x \).

8. **Identifying the Path**:
The equation derived is of the form \( y = ax - bx^2 \), which is the equation of a parabola. Thus, the path followed by the body is a parabolic trajectory.

**Hint**: Recall that parabolic motion is characteristic of projectile motion under uniform gravitational force.

### Conclusion:
The path followed by a moving body, when a constant force acts in a direction other than the initial velocity, is a **parabola**.