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!