An airport has runways only 184 m long. A

small plane must reach a ground speed of
40 m/s before it can become airborne.
What average acceleration must the plane’s
engines provide if it is to take off safely from
its airport?
Answer in units of m/s2

with the plane starting from rest and a uniform acceleration, the average velocity is 20 m/s

so the time of acceleration is 184/20 or 9.2 s

the acceleration is (40 m/s) / 9.2 s

To find the average acceleration, we can use the formula:

acceleration = (final velocity - initial velocity) / time

Since the plane needs to reach a ground speed of 40 m/s, and its initial velocity is 0 m/s (since it is at rest), the final velocity is 40 m/s.

Given that the distance traveled is 184 m, we need to find the time it takes to cover that distance before takeoff.

We can use the equation:

distance = (initial velocity * time) + (0.5 * acceleration * time^2)

Rearranging the equation to solve for time:

184 = 0.5 * acceleration * time^2

time^2 = (184 * 2) / acceleration

time = sqrt((368 / acceleration))

Now we can substitute the values into the first equation to find the average acceleration:

acceleration = (final velocity - initial velocity) / time
acceleration = (40 - 0) / sqrt((368 / acceleration))

To simplify the equation, we can square both sides:

acceleration^2 = 40^2 / (368 / acceleration)

Simplifying further:

acceleration^2 = 1600 / (368 / acceleration)

Multiplying both sides by (368 / acceleration):

acceleration * (368 / acceleration)^2 = 1600

Cancelling out the acceleration terms:

368^2 / acceleration = 1600

Rearranging the equation to solve for acceleration:

acceleration = 368^2 / 1600

Using a calculator:

acceleration ≈ 84.65625 m/s^2

Therefore, the average acceleration the plane's engines must provide for a safe takeoff is approximately 84.65625 m/s^2.

To find the average acceleration the plane's engines must provide, we can use the equation:

acceleration = (final velocity - initial velocity) / time

In this case, the plane must reach a ground speed of 40 m/s (final velocity) and it starts from rest (initial velocity = 0 m/s).

The time it takes for the plane to reach that speed can be calculated using the equation:

distance = (initial velocity + final velocity) / 2 * time

In this case, the distance is the length of the runway, which is given as 184 m. Using the given values, we can rearrange the equation to solve for time:

time = (2 * distance) / (initial velocity + final velocity)

Plugging in the values:

time = (2 * 184 m) / (0 m/s + 40 m/s)
time = 368 m / 40 m/s
time = 9.2 s

Now that we know the time it takes for the plane to reach the required speed, we can calculate the average acceleration:

acceleration = (final velocity - initial velocity) / time
acceleration = (40 m/s - 0 m/s) / 9.2 s
acceleration = 4.35 m/s^2

Therefore, the average acceleration the plane's engines must provide to take off safely from the airport is 4.35 m/s^2.