a projectile is thrown with speed u into air from a point on the horizontal ground at an angle `theta` with horizontal. If the air exerts a constant horizontal resistive force on the projectill then select correct alternative(s).

Options

A. At the farthest point, the velocity is horizontal
B. The time for ascent equals the time for descent
C. The path of the projectile may be parabolic
D. The path of the projectile may be a straight line.

Explanation

To solve the problem of a projectile thrown with speed \( u \) at an angle \( \theta \) with a constant horizontal resistive force acting on it, we can follow these steps:

### Step 1: Understand the Forces Acting on the Projectile
When the projectile is thrown, it experiences two main forces:
1. The gravitational force acting downward, which causes a downward acceleration \( g \).
2. A constant horizontal resistive force due to air resistance, which acts opposite to the direction of the projectile's horizontal motion.

### Step 2: Break Down the Motion into Components
The motion of the projectile can be analyzed in two dimensions:
- **Horizontal Motion**: The horizontal component of the initial velocity is \( u \cos(\theta) \). The horizontal acceleration \( a_x \) due to air resistance is negative (opposite to the direction of motion).
- **Vertical Motion**: The vertical component of the initial velocity is \( u \sin(\theta) \). The vertical acceleration is \( -g \) (downward).

### Step 3: Analyze the Horizontal Motion
The horizontal motion can be described by the equation:
\[ x(t) = u \cos(\theta) t - \frac{1}{2} a_x t^2 \]
where \( a_x \) is the constant horizontal resistive acceleration.

### Step 4: Analyze the Vertical Motion
The vertical motion can be described by the equation:
\[ y(t) = u \sin(\theta) t - \frac{1}{2} g t^2 \]

### Step 5: Determine the Path of the Projectile
Since the horizontal motion has a constant resistive force, the horizontal velocity will decrease over time, while the vertical motion is influenced only by gravity. The combination of these two motions results in a path that is not a simple parabola but rather a more complex trajectory that can be approximated as parabolic under certain conditions.

### Step 6: Conclusion
Given that the horizontal acceleration is constant and the vertical acceleration is also constant, the overall path of the projectile will be a combination of these two effects. Therefore, the path of the projectile may be parabolic or straight line depending on the values of \( u \), \( \theta \), and the magnitude of the resistive force.

### Final Answer:
The correct options are:
- C: The path of the projectile may be parabolic.
- D: The path of the projectile may be a straight line.

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