Question

A plane flies in a circle of radius 4,000 meters. The plane completes the circle in 2.5 minutes. What is the centripetal acceleration of the plane?

Answer 1

[AI-generated answer — Explain Bot]

To find the centripetal acceleration of the plane, we can use the formula:

𝑎 = 𝑣² / 𝑟

where 𝑎 is the centripetal acceleration, 𝑣 is the velocity of the plane, and 𝑟 is the radius of the circular path.

To find the velocity of the plane, we first need to calculate the circumference of the circle using the formula:

𝑐 = 2𝜋𝑟

where 𝑐 is the circumference of the circle.

Plugging in the radius of 4,000 meters, we have:

𝑐 = 2𝜋(4,000)

Simplifying, we find:

𝑐 ≈ 2𝜋(4,000) ≈ 8𝜋(1,000)

Now, we can use the formula for velocity:

𝑣 = 𝑐 / 𝑡

where 𝑡 is the time taken to complete the circle.

Plugging in the circumference and the time taken of 2.5 minutes (or 2.5 x 60 seconds), we have:

𝑣 = 8𝜋(1,000) / (2.5 x 60)

Simplifying further, we find:

𝑣 ≈ (8𝜋(1,000)) / 150

Now, we can plug this value of 𝑣 and the radius 𝑟 into the formula of centripetal acceleration:

𝑎 = (𝑣²) / 𝑟

Substituting the values:

𝑎 = ((8𝜋(1,000) / 150)²) / 4,000

Simplifying, we find:

𝑎 ≈ ((64𝜋²(1,000²)) / (150²)) / 4,000

Calculating further, we have:

𝑎 ≈ (64𝜋²(1,000²)) / (150² x 4,000)

Finally, multiplying and simplifying, we find:

𝑎 ≈ (64𝜋²(1,000²)) / (9 x 10⁸)

So, the centripetal acceleration of the plane is approximately:

𝑎 ≈ (64𝜋²(1,000²)) / (9 x 10⁸) ≈ 0.2808 m/s²

Therefore, the centripetal acceleration of the plane is approximately 0.2808 m/s².