*In terms of u, land g, obtain expressions for: the time t for which the motorbike is in the air*

To obtain the time \( t \) for which a motorbike is in the air, we need to look at the motion in the vertical direction. When an object (such as a motorbike) jumps off the ground or ramps into the air, its vertical motion follows the equations of uniformly accelerated motion under gravity.

Let's assume the following:

- \( u \) is the initial vertical velocity (which can be the vertical component of the motorbike's speed if it was launched at an angle).
- \( g \) is the acceleration due to gravity (approximately \( 9.8 \, \text{m/s}^2 \) on Earth).
- \( t \) is the time the motorbike is in the air.
- The displacement in the vertical direction is \( 0 \) (because the motorbike lands at the same vertical position from which it started).

We can use the following kinematic equation for vertical motion under gravity:

\[
y = u t - \frac{1}{2} g t^2
\]

For the time of flight, the motorbike returns to the same height it was launched from, so the displacement \( y \) is 0:

\[
0 = u t - \frac{1}{2} g t^2
\]

This is a quadratic equation in terms of \( t \). We can factorize it:

\[
t \left( u - \frac{1}{2} g t \right) = 0
\]

This gives two solutions:

1. \( t = 0 \) (the time when the motorbike is at the initial position)
2. \( t = \frac{2u}{g} \) (the time when it returns to the ground)

Thus, the total time the motorbike is in the air is:

\[
t = \frac{2u}{g}
\]

### Summary:
- The time \( t \) the motorbike is in the air is \( \frac{2u}{g} \), where \( u \) is the initial vertical velocity, and \( g \) is the acceleration due to gravity.

Would you like more details on the derivation or have additional questions?

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### Related Questions:
1. How do you calculate the horizontal distance traveled by the motorbike while in the air?
2. What happens to the time of flight if the initial velocity changes?
3. How can the angle of launch affect the time the motorbike is in the air?
4. How would air resistance change the calculation of the time in the air?
5. What other factors might influence the time in the air besides initial velocity and gravity?

**Tip:** To solve similar problems, always consider the vertical and horizontal components of motion separately.

In terms of u, land g, obtain expressions for: the time t for which the motorbike is in the air

Learn how to calculate the time a motorbike is in the air based on its initial vertical velocity (u) and gravity (g). This problem involves kinematic equations, focusing on uniformly accelerated motion, and is suitable for students in Grades 9-12.

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