A plane accelerates from rest at a constant rate of 5.00  m/s2
  along a runway that is 1800  m long. Assume that the plane reaches the required takeoff velocity at the end of the runway. What percentage of the takeoff velocity did the plane gain when it reached the midpoint of the runway?

To solve this problem, we need to use the kinematic equations of motion. Let's break it down step by step.

### Given:
- Acceleration, \( a = 5.00 \, \text{m/s}^2 \)
- Length of the runway, \( s = 1800 \, \text{m} \)

### 1. Calculate the takeoff velocity \( v \) at the end of the runway.

The kinematic equation that relates the final velocity \( v \), initial velocity \( u \), acceleration \( a \), and distance \( s \) is:

\[
v^2 = u^2 + 2as
\]

Since the plane starts from rest, \( u = 0 \). So:

\[
v^2 = 2as
\]

Substituting the given values:

\[
v^2 = 2 \times 5.00 \, \text{m/s}^2 \times 1800 \, \text{m}
\]

\[
v^2 = 18000 \, \text{m}^2/\text{s}^2
\]

\[
v = \sqrt{18000} \, \text{m/s}
\]

\[
v \approx 134.16 \, \text{m/s}
\]

So, the takeoff velocity at the end of the runway is approximately \( 134.16 \, \text{m/s} \).

### 2. Calculate the velocity at the midpoint of the runway.

The midpoint of the runway is at \( s = 900 \, \text{m} \).

Using the same kinematic equation:

\[
v_{\text{mid}}^2 = 2as_{\text{mid}}
\]

\[
v_{\text{mid}}^2 = 2 \times 5.00 \, \text{m/s}^2 \times 900 \, \text{m}
\]

\[
v_{\text{mid}}^2 = 9000 \, \text{m}^2/\text{s}^2
\]

\[
v_{\text{mid}} = \sqrt{9000} \, \text{m/s}
\]

\[
v_{\text{mid}} \approx 94.87 \, \text{m/s}
\]

So, the velocity at the midpoint of the runway is approximately \( 94.87 \, \text{m/s} \).

### 3. Calculate the percentage of the takeoff velocity at the midpoint.

\[
\text{Percentage} = \left(\frac{v_{\text{mid}}}{v}\right) \times 100\%
\]

Substituting the values:

\[
\text{Percentage} = \left(\frac{94.87 \, \text{m/s}}{134.16 \, \text{m/s}}\right) \times 100\%
\]

\[
\text{Percentage} \approx 70.71\%
\]

So, the plane gained approximately **70.71%** of the takeoff velocity when it reached the midpoint of the runway.

Would you like more details or have any questions? Here are five related questions for further practice:

1. What is the time taken for the plane to reach the midpoint of the runway?
2. What is the time taken for the plane to reach the end of the runway?
3. What is the distance covered by the plane in the first 5 seconds?
4. If the acceleration were doubled, what would be the new takeoff velocity?
5. How much time would the plane save if the runway were only 900 meters long with the same acceleration?

**Tip:** Understanding the relationship between acceleration, velocity, and distance is crucial for solving kinematic problems efficiently.

Explore how to calculate the percentage of takeoff velocity gained by a plane when reaching the midpoint of a runway. Understand the application of kinematic equations and acceleration in solving this physics problem.

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