To solve this problem, we will use the basic kinematic equations for constant acceleration. Let's break it down step by step for both parts (a) and (b).

Given Data:

• Distance covered: $d = 8.27 \, \text{km} = 8270 \, \text{m}$
• Time: $t = 31.1 \, \text{s}$
• Constant acceleration: $a = 3.82 \, \text{m/s}^2$

Part (a): Initial Speed $v_0$

We need to find the initial speed $v_0$ of the airplane at the start of the time period. For this, we can use the following kinematic equation:

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

Where:

• $d$ is the distance,
• $v_0$ is the initial speed (what we need to find),
• $t$ is the time,
• $a$ is the acceleration.

Rearranging the equation to solve for $v_0$:

$v_0 = \frac{d - \frac{1}{2} a t^2}{t}$

Now, substituting the known values:

$v_0 = \frac{8270 - \frac{1}{2} \times 3.82 \times (31.1)^2}{31.1}$

Let me calculate that for you.

Part (b): Final Speed $v_f$

To find the final speed $v_f$ of the airplane at the end of the 31.1 seconds, we can use another kinematic equation:

$v_f = v_0 + a t$

After calculating $v_0$ from part (a), we will substitute it into this equation along with the known acceleration and time to find the final speed.

Let me calculate both $v_0$ and $v_f$.### Results:

• Initial speed of the airplane at the beginning of 31.1 seconds: $v_0 = 206.52 \, \text{m/s}$ (rounded to two decimal places).
• Final speed of the airplane at the end of 31.1 seconds: $v_f = 325.32 \, \text{m/s}$ (rounded to two decimal places).

Would you like more details on the calculations or explanations?

Here are some related questions to explore:

1. What is the total distance covered by the airplane if the acceleration continues for another 10 seconds?
2. How long would it take the airplane to reach a speed of 500 m/s under the same acceleration?
3. What would be the airplane's speed after 20 seconds if it started from rest?
4. How would the result change if the acceleration were reduced to 2 m/s²?
5. What would happen to the distance covered if the acceleration were doubled?

Tip: For constant acceleration, using the right kinematic equation helps solve for unknowns like speed, time, or distance based on what is provided.