The question aims to find if a **plane** can land on a **small tropical island** if the runway is **shorter** than a **kilometer.**

The question depends on the concept of **3rd equation** of **motion.** The **3rd equation** of **motion** yields **final velocity** given a **uniform acceleration** and **initial velocity** over a given **distance.** The formula for **3rd equation** of **motion** is given as:

\[ v_f^2 = v_i^2 + 2 a S \]

$v_i$ is the specific **initial velocity** of the object.

$v_f$ is the specific **final velocity** of the object.

$a$ is the **uniform acceleration** of the object.

$S$ is the **distance** traveled by the object.

## Expert Answer

In this question, we are given some information about a jet plane that needs to **land** on a **small tropical island.** Our objective is to find out whether the plane will be to make a **successful landing** on the **runway** or not. The information that was given about the problem is as follows:

\[ Initial\ Velocity\ of\ the\ Plane\ v_i = 100\ m/s \]

\[ Uniform\ Acceleration\ of\ the\ Plane\ a = – 7\ m/s^2 \]

\[ Distance\ of\ the\ Runway\ S = 0.900\ km \]

As the **plane** needs to be **fully stopped** at the end of the **runway,** the **final velocity** of the plane is given as:

\[ Final\ Velocity\ of\ the\ Plane\ v_f = 0\ m/s \]

We need to determine whether the **plane** will be available to **land** on the runway or not. So we need to calculate the **distance** the plane would travel to **fully stop** given this information.

As we have both the **initial** and **final velocities** of the plane with its **uniform acceleration,** we can use the **3rd equation** of **motion** to calculate the **distance** for the plane. One thing of note here is that we do not have the **value** of **time** for the jet plane, so we cannot use the **2nd equation** of **motion,** which uses time. The **3rd equation** to motion is given as:

\[ v_f^2 = v_i^2 + 2 a S \]

Substituting the values, we get:

\[ (0)^2 = (100)^2 + 2 \times – 7 \times S \]

Rearranging the values to calculate the **distance.**

\[ S = \dfrac{ (100)^2 }{ 2 \times 7 } \]

\[ S = \dfrac{ 10000 }{ 14 } \]

\[ S = 714.3\ m \]

\[ S = 0.714\ km \]

The **runway** is **0.900 km long,** and the **jet plane** needs about **0.714 km** to **fully stop** after **landing.** So the jet plane will be able to **successfully land** on the **small tropical island.**

## Numerical Results

The **distance** needed for the **jet plane** to land is about **0.714 km**, while the **runway** is **0.900** **km** long. The **jet plane** will be able to land on the small tropical island.

## Example

An **airplane** has an **initial** velocity of **150 m/s** with an **acceleration** of $5 m/s^2$. It needs to land a runway in the **Himalayas mountains,** but the runway is only **800** **m long.** Can this **airplane land** at the airport situated high in the mountains?

Given the information, we can use the **3rd equation** of **motion** to calculate the **distance** the airplane will take to stop.

\[ v_f^2 = v_i^2 + 2 a S \]

Substituting the values, we get:

\[ S = \dfrac{ 150^2 }{ 2 \times 5 } \]

\[ S = \dfrac{ 22500 }{ 10 } \]

\[ S = 2250 m \]

The **airplane** needs a **2250** **m** long runway to **stop,** so it will **not** be able to **land** at the **airport** in the **mountains.**