3.	A certain airplane has an acceleration of 15.0 m/s2.
a)	How fast will it be moving 2.5 s after it starts down the runway?
b)	How far down the runway will it travel during the 2.5 s?
c)	Minimum take-off speed is 60.0 m/s. How long must the runway be?

We are given that the airplane has an acceleration of \( a = 15.0 \, \text{m/s}^2 \), and we need to find:

### a) How fast will it be moving 2.5 s after it starts down the runway?
We can use the equation for velocity under constant acceleration:
\[
v = u + at
\]
Where:
- \( v \) is the final velocity
- \( u \) is the initial velocity (since it starts from rest, \( u = 0 \))
- \( a = 15.0 \, \text{m/s}^2 \) is the acceleration
- \( t = 2.5 \, \text{seconds} \)

Substituting the values:
\[
v = 0 + (15.0 \, \text{m/s}^2)(2.5 \, \text{s}) = 37.5 \, \text{m/s}
\]
So, the airplane will be moving at **37.5 m/s** after 2.5 seconds.

### b) How far down the runway will it travel during the 2.5 s?
For distance, we use the equation:
\[
s = ut + \frac{1}{2}at^2
\]
Where:
- \( s \) is the distance traveled
- \( u = 0 \, \text{m/s} \)
- \( a = 15.0 \, \text{m/s}^2 \)
- \( t = 2.5 \, \text{seconds} \)

Substituting the values:
\[
s = (0)(2.5) + \frac{1}{2}(15.0)(2.5)^2 = \frac{1}{2}(15.0)(6.25) = 46.875 \, \text{m}
\]
So, the airplane will travel **46.9 meters** down the runway in 2.5 seconds.

### c) How long must the runway be if the minimum take-off speed is 60.0 m/s?
We use the same velocity equation:
\[
v = u + at
\]
Rearranging it to solve for time \( t \):
\[
t = \frac{v - u}{a} = \frac{60.0 - 0}{15.0} = 4.0 \, \text{seconds}
\]
Now, to find the required runway length, use the distance formula:
\[
s = ut + \frac{1}{2}at^2
\]
Substitute the values:
\[
s = (0)(4.0) + \frac{1}{2}(15.0)(4.0)^2 = \frac{1}{2}(15.0)(16) = 120 \, \text{m}
\]
Thus, the minimum runway length must be **120 meters**.

### Summary:
a) Final speed after 2.5 s: **37.5 m/s**  
b) Distance traveled in 2.5 s: **46.9 m**  
c) Minimum runway length for take-off: **120 m**

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. What happens to the take-off time if the acceleration is increased?
2. How would the distance traveled change if the airplane accelerated for 5 seconds?
3. What role does air resistance play in the actual take-off process?
4. How would you calculate the energy required for take-off?
5. Can you determine the force acting on the airplane during acceleration?

**Tip:** Always double-check the units in your calculations to ensure consistency!

A certain airplane has an acceleration of 15.0 m/s^2.
a) How fast will it be moving 2.5 s after it starts down the runway?
b) How far down the runway will it travel during the 2.5 s?
c) Minimum take-off speed is 60.0 m/s. How long must the runway be?

Explore how to calculate the speed, distance traveled, and runway length for an airplane accelerating at 15.0 m/s^2. This problem uses fundamental kinematics equations to determine the velocity after 2.5 seconds, the distance traveled, and the minimum runway length for take-off at 60.0 m/s. Suitable for high school students.

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