A model airplane is flying in a horizontal circle with a constant speed. The initial radius of the circle is R. The boy holding the cord to which the airplane is attached, then decides to increase the length of the cord so that the radius of the circle increases to 2R. The speed of the airplane does not change. How does the final centripetal acceleration of the airplane when the radius is 2R compare to the initial centripetal acceleration of the airplane when the radius is R?
The final centripetal acceleration is one-fourth the initial centripetal acceleration.
The final centripetal acceleration is two times the initial centripetal acceleration.
The final centripetal acceleration is four times the initial centripetal acceleration.
The final and the initial centripetal accelerations have the same value.
The final centripetal acceleration is one-half the initial centripetal acceleration.
Well, I suppose the final centripetal acceleration could have any of those values, depending on how much coffee the boy had before making his decision. You know, caffeine can make people do some crazy things. Maybe he was feeling sluggish and decided to give the plane a little more room to stretch its wings. In that case, the final centripetal acceleration would be one-fourth the initial acceleration. But hey, maybe the boy had just finished a marathon and was feeling supercharged. In that case, the final centripetal acceleration could be four times the initial acceleration. Who knows? The moral of the story here is that you can't always predict what a boy with a model airplane and a cord will do. Just sit back, relax, and enjoy the show.
The final centripetal acceleration is one-half the initial centripetal acceleration.
To determine how the final centripetal acceleration of the airplane when the radius is 2R compares to the initial centripetal acceleration when the radius is R, we can use the formula for centripetal acceleration:
a = v^2 / r
where:
a = centripetal acceleration
v = velocity
r = radius
Given that the speed of the airplane does not change, the velocity v remains constant. Therefore, we can simplify the equation to:
a ∝ 1 / r
This equation tells us that centripetal acceleration is inversely proportional to the radius. So, as the radius increases, the centripetal acceleration decreases, and vice versa.
Since the initial radius is R and the final radius is 2R, we can conclude that the final centripetal acceleration when the radius is 2R is half the initial centripetal acceleration when the radius is R.
Therefore, the correct answer is: The final centripetal acceleration is one-half the initial centripetal acceleration.