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The problem asks to determine the distance traveled by an airplane accelerating down a runway at \( 3.20 \, \text{m/s}^2 \) for \( 32.8 \, \text{s} \) before takeoff.

We can solve this using the kinematic equation for uniformly accelerated motion:

\[
d = v_0 t + \frac{1}{2} a t^2
\]

Where:
- \( d \) is the distance traveled (what we're solving for),
- \( v_0 \) is the initial velocity (since the airplane starts from rest, \( v_0 = 0 \)),
- \( a = 3.20 \, \text{m/s}^2 \) is the acceleration,
- \( t = 32.8 \, \text{s} \) is the time.

Since \( v_0 = 0 \), the equation simplifies to:

\[
d = \frac{1}{2} a t^2
\]

Now, substituting the given values:

\[
d = \frac{1}{2} (3.20 \, \text{m/s}^2) (32.8 \, \text{s})^2
\]

Let's calculate this:The distance traveled by the airplane before takeoff is approximately \( 1721.34 \, \text{meters} \).

Would you like any additional details or have further questions? Here are a few related problems for you to explore:

1. How long would the plane need to accelerate if the runway was only 1500 meters long?
2. What would the final velocity of the airplane be after 32.8 seconds?
3. If the plane had an initial velocity of 10 m/s, how would that affect the distance calculation?
4. How much time would it take to travel 2000 meters at the same acceleration?
5. How does changing the acceleration to \( 4 \, \text{m/s}^2 \) affect the distance traveled?

**Tip:** When solving kinematic problems, identifying whether initial velocity is zero simplifies the use of motion equations.

An airplane accelerates down a runway at 3.20 m/s² for 32.8 s until it finally lifts off the ground. Determine the distance traveled before takeoff.

This example problem demonstrates how to calculate the distance an airplane travels while accelerating at 3.20 m/s² for 32.8 seconds using kinematic equations. Suitable for high school students (Grades 9-12), this solution covers key concepts in physics, including uniformly accelerated motion and the kinematic equation d = 1/2at².

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