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

Aha!

The answer is e.
Anyone who said that the answer is "none of the above" is wrong.
Why?

You can't apply the equations of circular motion to this problem!

Why? The condition for circular motion is that the force applied must always be directed towards the centre of the circular path it is following.
In this problem, this is not satisfied. Draw force diagrams if you don't see it.

Fortunately, there's still a very easy way to solve this:

All you need to use is the fact that velocity is a vector.

Then you use

acceleration = change in velocity / time

Now if the velocity changes from 500km/h north to 500km/h south (or -500km/h north if you prefer), the total change in velocity is 1000km/h.

so
acceleration = 1000 / 40 = 25 km/h per second

Now, about the direction.
Initially, to turn the plane, the force applied must be directed east. At the end of the motion, it must be directed west. At the mid-point it must have been directed south.
The east and west components cancel, so the average acceleration (which is proportional to the force) is directed south.

I hope this helps you out! Feel free to email me if you need more help.

Answer 2

I agree with kirchwey. I think the answer is "none of the above": 39.27 km/h.s south.

From the beginning of the turn to the end of the turn, the acceleration is constant at this value, and so the average acceleration is this value. The acceleration vector is constant in length, and always points off the right wing of the airplane, to the center of the circle.

The component of the acceleration in the north-south direction varies as [-39.27 km/h.s]*sin(t*pi/[40 sec]). The time-average of this quantity is its integral from t = 0 to t = 40 sec, divided by 40 sec, which is -25 km/h.s (negative, in this case, meaning "pointing south"). But the problem did not ask for "the average of the component of acceleration in the N-S direction", it asked for "the average acceleration". The problem is faulty.

To put it another way, if the pilot does a "coordinated turn", then during the turn the passengers in the airplane will feel a constant acceleration of 14.67 m/s^2 downward into their seats during the turn, that results from the vector sum of the constant gravitational acceleration vector plus the constant (perpendicular) acceleration of the turn. So the passengers feel the turn as a steady, 1.5 g acceleration.

Answer 3

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.

Answer 4

You are doing 500kph north , you apply a constant acceleration and 40 seconds later you are heading at the same speed South.
Your change in velocity was 1000kph in 40 seconds so d or e. Neither of these give east as a plausible direction for your average acceleration and they certainly shouldn't as you started off going 0kph east and finished up doing the same speed east - no average acceleration. So just choose; was the average acceleration towards the north or south? d or e?

Answer 5

The key point is that you're looking for the AVERAGE acceleration. This allows you to use formula:
a(av) = deltaV/deltaT
= (V(final) - V(initial)) / deltaT

Okay, slap a coordinate system on the diagram so that North = positive and South = negative.

Then,
a = (-500 -500 )/ 40
= -25 km/hour/second.

This is 25 south.

The extraneous information about tangential and radial acceleration was just put there to see if you knew the difference between average and instantaneous.

Answer 6

centripetal stress is responsible for the action of a body in a round route. centripetal stress is defined because the web stress appearing on the body in the route of THE CENTRE, accordingly contained in the case given through you, the needed centripetal stress is geared up through mg i.e weight and stress. both those forces are only responsible for providing the centripetal stress to the body. accordingly the web stress in the route of the centre would take delivery of through mg + t. in spite of the indisputable fact that the price of centripetal stress is discovered to be mv^2/r. accordingly we are able to write mg + t = mv^2/r