Jet fly by
A farmer on an open tractor is traveling on a straight road heading north at 1 0 h r k m . A jet pilot is training by flying at high speed and low altitude in the area. The jet is travelling at a steady speed of 8 0 0 h r k m at an altitude of 100 meters when it crosses the road the tractor is traveling on. The jet is traveling west in a straight line. The plane crosses the road 100 meters south of the tractor's position on the road. Because of the noisy tractor the farmer cannot hear the approaching plane coming until the plane is 300 meters or less from the tractor. What is the plane's position, expressed in meters, east of the road when the farmer first hears the plane?
Assume the speed of sound is 3 4 0 s m
The answer is 68.79.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
2 solutions
The original answer I submitted for this question was 69.7 m. That answer is not correct as it did not consider all of the factors correctly. The tractors movement while the sound was traveling to the farmer was not factored into the equation. i.e. 0.882 times 2.7778.. That answer would be correct if the sound was heard by the farmer the instant the plane was 300 m away with no consideration for the travel time of the sound.
The solution submitted by G. Butel was 68.496 which is also incorrect as it does not consider the tractors moment correctly. That answer would be correct if the tractor was not moving at all i.e. it was just sitting 100m north of the planes crossing point.
clearly nether of those solutions are correct. I have edited my solution to consider the tractor movement.
I believe the correct answer to be 68.790 m when considering all of the information given
For accuracy, I have a slight concern about what it means for "until the plane is 300 meters or less from the tractor". Is this distance measured as the "then plane" position to the "now tractor" position, accounting for the time it takes sound to travel?
If so, this question would be more complicated, and we should be using an inertial frame of reference instead.
Log in to reply
My intention for this problem was based on how far the sound traveled from the source to the farmer ear. The sound of the jet must originate from spot 300 m or less from the point where the farmer is when he can first hear it. Since the perceived volume of a sound, hence the ability to hear that sound, is inversely proportional to the distance the sound has traveled. I intended this problem to mean --if the sound from the jet had traveled more then 300 meters to the farmer ear he would not hear it
The problem I have with my original solution was, it found the point when the plan was 300 m from the tractor. The tractor continues to move as the sound comes toward it. by the time the sound reaches the tractor, the tractor has moved north a small amount. this movement would put the travel distance of the sound outside the 300 m restriction given in the problem. My current solution takes this tractor movement into account, so that as the sound reaches the farms ear it has traveled 300 m from the plane when the sound was produced.
My approach to this change is, since we know how long the sound has been coming we can determine how far the tractor has moved in that time. The effect would be the plane would be closer to the road then my original solution because we are taking into account that the sound must travel that distance further north while still traveling a total distance of 300m..
Log in to reply
I have updated the answer to 68.79.
As we allow for a 2% error margin, those who answered 69.7 are still considered correct.
Log in to reply
@Calvin Lin – Great, . 2% is very generous for a question like this. The change I made to the approach to a solution makes a small difference in this problem. If the problem was changed to consider a faster moving vehicle on the road or possibly two planes at altitude the change would become very significant..
Log in to reply
@Darryl Dennis – The 2% error margin was introduced to help questions that require inputing values for constants, especially in Physics. If you think that you want a much finer error margin, an approach would be to ask for ⌊ 1 0 S ⌋ , in which case the correct answer would be for 6 8 . 7 ≤ S < 6 8 . 8 .
Let me know if you would prefer this, and I can update the question accordingly.
Ground distance from Jet to tractor in order that sound be heard = Sqrt(300^2 - 100^2) m.
I took 100m for the north component of the distance the tractor is from the jet at the point the sound could be heard and obtained 68.496 m for the final answer.
I was not completely satisfied with my analysis of the problem because the tractor was not 100 m North at that point in time however I submitted this answer.
I was surprised that this was claimed correct by the computer program even though the answer was given as 69.7.
Why???
On further consideration I found that the N component was 96.67742 m, the E component was 265.8072172 and the final answer is 69.729
Can you explain how 69.7?
Log in to reply
Do you understand that the N component of the tractor at the time when the distance between tractor and jet was 300 m, is less than 100 m?
Log in to reply
I got the N and E component exactly as yu.
Log in to reply
@Anitus Raj – It isn't 100 m for the N component. It's less.
Log in to reply
@Guiseppi Butel – Yes ,the N component s less than 100.
Please see the discussion on Darryl's solution. This question is made intriguing by the fact that it takes time for sound to travel.
Because we allow for a 2% margin of error, your answer of 68.5 was close enough to the (then) answer of 69.7.
Tractor speed = 10 km/hr 10 * 1000/3600 = 2.77778m/s
Plane speed =800 km/hr =222.22 m/s
Let time 0 be the point when the plane is crossing the road 100m south of the tractor.
The distance from the plane to the tractor can be expressed as
D=/sqrt{(horizontal distance)^2 + (north distance)^2 + (west distance)^2)}
Distance at time T seconds before the plane reaches the road.
Horizontal dis =100: North dis =(100 - 2.777 T): west dis =(222.2 T)
Time for sound to reach tractor/farmer = 300/340 seconds
The distance north that sound must travel is the distance north to the tractor at time T plus the distance north the tractor moves while the sound in traveling
North dis = 100 - (2.778(T - 0.882))
What is time T when the plane and the tractor are 300m apart? When the farmer can first hear the plane.
Sqrt( 100^2 + (100 + 2.450 - 2.778 T )^2+ (222.2 T) ^2) = 300.
Using (a -b)^2 = a^2 + b^2 - 2 then the quadratic equation to find T
T = 1.192 seconds
Planes position west when it is 300 m from tractor =1.192 \times 222.2 = 264.89 m
Time of sound travel is 300 / 340 = .882 sec
Plane travel west as sound travels to tractor. 882 times 222.2 = 196.100 m
Planes position as sound reaches farmer 264.89 - 196.1 = 68.79 m