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This problem involves a motorbike that is launched from a symmetrical ramp and becomes airborne. We are tasked with finding the time of flight and the horizontal distance traveled by the motorbike on the right side of the ramp. We can solve this problem using the principles of projectile motion.

### Given Data:
- Initial speed: \( u \)
- Gravitational acceleration: \( g \)
- The ramp is symmetrical with angle \( \theta \).
- Distance along the ramp: \( l \)
  
We need to find:
1. **The time \( t \) for which the motorbike is in the air.**
2. **The horizontal distance \( OP = l \) (distance along the ramp).**

### Part (a): Time in the Air

At the top of the ramp, the motorbike is launched with a velocity \( u \) at an angle \( \theta \) above the horizontal. We need to determine the time the motorbike stays airborne (its flight time).

Using the vertical component of the velocity:
- The vertical component of the initial velocity is \( u \sin \theta \).
- The total time of flight \( t \) in projectile motion is given by:
  \[
  t = \frac{2u \sin \theta}{g}
  \]
This formula arises because the time to reach the maximum height is \( \frac{u \sin \theta}{g} \), and the total time of flight is twice this time (as the ascent and descent times are equal).

### Part (b): Distance OP (Horizontal Distance)

The horizontal component of the velocity is \( u \cos \theta \), and the horizontal distance traveled during the flight is given by:
\[
OP = \text{Horizontal velocity} \times \text{Time of flight}
\]
Substituting the known values:
\[
OP = u \cos \theta \times \frac{2u \sin \theta}{g}
\]
\[
OP = \frac{2u^2 \sin \theta \cos \theta}{g}
\]
Using the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \), we get:
\[
OP = \frac{u^2 \sin(2\theta)}{g}
\]

Thus, the horizontal distance OP is:
\[
OP = \frac{u^2 \sin(2\theta)}{g}
\]

---

Would you like further clarifications or details?

Here are some relative questions for practice:
1. What is the maximum height reached by the motorbike in this projectile motion?
2. How would the time of flight change if the angle of launch \( \theta \) was increased?
3. What would happen to the distance \( OP \) if the initial speed \( u \) was doubled?
4. How does the horizontal velocity change during the flight?
5. What would be the effect of air resistance on the time of flight and distance traveled?

**Tip:** When dealing with projectile motion, always break the motion into horizontal and vertical components and apply the respective kinematic equations.

A motorbike is ridden up the left side of a symmetrical ramp. The bike reaches the top of the ramp at speed u, becomes airborne, and falls to a point P on the other side of the ramp. In terms of u, l, and g, obtain expressions for: (a) the time t for which the motorbike is in the air, and (b) the distance OP (= l) along the right side of the ramp.

Solve a projectile motion problem involving a motorbike launched from a symmetrical ramp. Learn how to calculate the time of flight and the horizontal distance traveled using key concepts in kinematics and trigonometry. Suitable for students in Grades 11-12.

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