The time of flight of a projectile on an upwardd inctined plane depends upon

Options

A. angle of inclination of the plane
B. angle of projection
C. the value of acceleration due to gravity
D. all of these.

Answer

To determine the time of flight of a projectile on an upward inclined plane, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have a projectile launched at an angle \(\alpha\) with respect to the horizontal, and it lands on an inclined plane that makes an angle \(\theta\) with the horizontal.

2. **Identify the Components of Motion**:
- The initial velocity \(u\) can be broken down into two components:
- Horizontal component: \(u \cos \alpha\)
- Vertical component: \(u \sin \alpha\)

3. **Consider the Forces Acting on the Projectile**:
- The gravitational force acting on the projectile is \(mg\), which can be resolved into two components relative to the inclined plane:
- Perpendicular to the incline: \(mg \cos \theta\)
- Parallel to the incline: \(mg \sin \theta\)

4. **Determine the Acceleration**:
- The acceleration of the projectile along the inclined plane due to gravity is \(g \cos \theta\).

5. **Use the Equation of Motion**:
- The vertical displacement \(y\) can be expressed as:
\[ y = u \sin \alpha \cdot t - \frac{1}{2} g \cos \theta \cdot t^2 \]
- Since the projectile returns to the same vertical level when it hits the inclined plane, we set \(y = 0\):
\[ 0 = u \sin \alpha \cdot t - \frac{1}{2} g \cos \theta \cdot t^2 \]

6. **Solve for Time of Flight**:
- Rearranging the equation gives:
\[ u \sin \alpha \cdot t = \frac{1}{2} g \cos \theta \cdot t^2 \]
- Factoring out \(t\) (noting \(t \neq 0\)):
\[ t \left( u \sin \alpha - \frac{1}{2} g \cos \theta \cdot t \right) = 0 \]
- This leads to:
\[ t = \frac{2u \sin \alpha}{g \cos \theta} \]

7. **Identify Dependencies**:
- From the derived formula \(t = \frac{2u \sin \alpha}{g \cos \theta}\), we can see that the time of flight \(t\) depends on:
- The angle of projection \(\alpha\)
- The angle of inclination \(\theta\)
- The acceleration due to gravity \(g\)

### Conclusion:
The time of flight of a projectile on an upward inclined plane depends on the angle of projection \(\alpha\), the angle of inclination \(\theta\), and the gravitational acceleration \(g\). Therefore, the correct answer is that it depends on all of these factors.