Circular motion problem?

Question

An airplane is flying north at 500km/h. It makes a gradual 180degr. turn at constant speed, changing its direction of travel from north through east to south. The process takes 40s. The average acceleration of the plane for this turn in km/h.s is:
a 12.5 km/h.s, north
b 12.5 km/h.s, east
c 12.5 km/h.s, south
d 25 km/h.s, north
e 25 km/h.s, south


Ok, I know I have a=v^2/r and T=2pir/v to work with.
40 sec = 0.011 hr is half a revolution (180degr) and since T is a hole revolution I think T is 0.0222. Putting that into the equation, I get 1.768km for r.
I tried working the problem backwards by plugging in the answers but I just don't get it. (maybe I'm doing omething wrong with the units since a is normally m/s^2 and thesy give me the answer in km/h.s?)
Also the direction, I would say if you travel north and make a turn east and end up going south, the average direction is east, does that make sense?

Answer 1

Since the choice of answers includes a direction, it sounds to me like the question is asking not for the radial acceleration but the net velocity change divided by the time, or 1000 km/hr divided by 40 sec. Thus the answer = e.
Averaging the radial acceleration yields the same answer! It's easiest to use consistent time units for this, so change 500 km/hr to 0.1389 km/s. In 40 s the plane travels 5.556 km and pi radians for a radius of 5.556/pi, or 1.7685 km. Then radial acceleration v^2/r = 0.01091 km/s^2 or 39.27 (km/hr)/s. The acceleration starts in an easterly direction, and rotates through southerly to westerly. (If this isn't clear, visualize your path, and remember the plane accelerates to its right during the turn.) To get the average southerly component you multiply by 2/pi and the result is exactly 25 (km/hr)/s. All the east-west components cancel.
You asked this question twice. Not a great idea, since you get identical answers from opportunists like me!

Answer 2

If the units are throwing you off, just convert them. Remember, you have 1000m/1km and 3600s/1hr

Since you have a multiple choice question, we can reduce the number of answers by logic. Since the plane eventually points south, the average acceleration must be southerly. An easterly average acceleration would leave the plane pointing east, and a northerly acc. would simply increase the plane's speed.

Your second equation can be simplified, since we are only looking for the circumference of a semi-circle. It becomes simply t = (pi*r)/v

Plugging in values, (Your conversion is correct), and solving for R, we get 1.768 km for r. So far so good.

Here's the final hurdle. I'm betting this is where you ran into trouble. Everything that happens to numbers in an equation also happens to units. plugging 500km/hr into the v^2/r equation would give you an answer in km/hr^2. We need to convert the final answer.
I get 141x10^3, or approximately 141000 km/hr^2. If we multiply by the conversion factor, 1hr/3600s, we get about 40 km/(hr*s).

Even though this isn't exact, multiple choice strategies suggest picking the closest answer if you don't hit one exactly, which would be 25km/(h*s).

One other thing which may complicate things is the fact that with circular motion, the acceleration vector is constantly changing. The acceleration is toward the center of the circle.

Good Luck, hope this helps.