A heavy particle is projected with a velocity at an angle with the horizontal into the uniform gravitational field. The slope of the trajectory of the particle varies as

Options

A.
B.
C.
D.

Explanation

To solve the problem of how the slope of the trajectory of a heavy particle varies when projected at an angle with the horizontal, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a heavy particle projected with an initial velocity \( u \) at an angle \( \theta \) with the horizontal. We need to find out how the slope of the trajectory changes.

2. **Equation of Trajectory**:
The equation of the trajectory of a projectile in a uniform gravitational field is given by:
\[ y = x \tan \theta - \frac{g x^2}{2 u^2 \cos^2 \theta} \]
where:
- \( y \) is the height,
- \( x \) is the horizontal distance,
- \( g \) is the acceleration due to gravity,
- \( u \) is the initial velocity,
- \( \theta \) is the angle of projection.

3. **Finding the Slope**:
To find the slope of the trajectory, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \tan \theta - \frac{g}{u^2 \cos^2 \theta} x \]
This expression represents the slope of the trajectory at any point \( x \).

4. **Analyzing the Slope**:
The slope \( \frac{dy}{dx} \) is a linear function of \( x \). The term \( -\frac{g}{u^2 \cos^2 \theta} x \) indicates that the slope decreases as \( x \) increases, since it has a negative coefficient.

5. **Conclusion**:
Therefore, the slope of the trajectory varies linearly with \( x \) and decreases as \( x \) increases. This means that the slope is negative and changes in a straight line manner.

### Final Answer:
The slope of the trajectory of the particle varies linearly with \( x \) and is negative.

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