Snow Plow, Know How

• 1. ByPatrick McCoyWith assistance ByDr. Danrun Huang

2. Here in Minnesota snow plows are part of our every day life and seans such asthis are common every winter.http://www.youtube.com/watch?v=z0cMI_gVVaI The snow plow problem has been covered many times in differential equationsand is a staple many differential equations text books. Before I can discuss the new parts of the problem I will need to give you anover view of the old problem. The snow is coming down at a constant rate. As the snow gets deeper the plow goes slower. It is typically split in to two parts. Here is how the typical problem reads.(a)One morning it began to snow very hard and continued snowing steadilythroughout the day. A snowplow set out at 8:00 A.M. to clear a road, clearing 2 miby 11:00 A.M. and an additional mile by 1:00 P.M. At what time did it startsnowing? (b) One day it began to snow exactly at noon at a heavy and steady rate. Asnowplow left its garage at 1:00 P.M. and another one followed in its tracks at2:00 P.M. At what time did the second snowplow crash into the first? Could thecrash have been avoided by dispatching the second snowplow at a later time? 3. We want to know when it started snowing. Time is measured in hours x=x(t)= the distance the plow has traveled in t hours x(0)=0 x(3)=2 x(5)=3 toc=0 k = an unknown constant volume that the plow can remove perunit time r=the rate of falling snow per unit volume per unit time 7. h2(t,y)=h2 isdependent how long it has beensince the first plow cleared the road. So at some future time lets say time T x(T)=Aln(T)=y Solving So h2(t,y)=r(t-T) Then Using the inverse Thisis simply a first order differentialequation 8. Solving this first order Differential equationgives When t=2, y=0 so C=2 This give the equation If we substitute our equation x(t)=A ln(t)=yinto this equation we get : So they crash at 2:43 P.M. 9. Wecan use B.1 to solve B.2 We already have an equation for t This time lets leave constant c in place Again substituting x(t)=A ln(t)=y into thisequation we get : Nomatter what a crash occurs because t is never less than 0 10. The problem One day it began to snow exactly at noon at a heavy and steady rate.Snowplow #1 left its garage at 1:00 p.m. and moved east. After awhile Snowplow #2 left a different garage and moved south. Supposethat the snowplows and the steady rate of snow are the same as thatin Problem 1, the garage of Snowplow #1 is 4 miles from theintersection and the Garage of Snowplow #2 is 3 miles from theintersection. (a) The two snowplows crash into each other at the intersection of twoperpendicular roads. At what time did Snowplow #2 leave its garage? (b) If snowplow #1 left its Garage at 2 p.m. and the two plows crash at theintersection, at what time did Snowplow 32 leave its Garage? Why? (c) Snowplow #1 left its garage at 1 p.m. and Snowplow #2 left its garage at1:15p.m. There is nothing to prevent the two plows from seeing each otherexcept the snow. Suppose the visibility of that day is only 1/10 miles. Couldthe drivers on the two plows see each other? 11. At what time did Snowplow #2 leave itsgarage? t=0 at noon when it started snowing x=x(t)= the displacement of the first plow x(1)=0 Y=y(t)= the displacement of the second plow From the first problem we already knowA=5.1295 12. For the first plow similar to problem one The crash happens when For plow two we have So 13. Now we know that in order for the two plows to collide the two plows must have been the same distance from the intersection when the second plow started out. This leads two possible ways to solve the problem. 14. Firstly if they were the same distance from theintersection when plow two started out due tothe steady rate of snow fall then So Alternatively I could have solved it this way We also know that So the second plow left 1.22 hours after itstarted snowing or 1:12 p.m. 15. At what time did Snowplow #2 leave itsgarage? t=0 at noon when it started snowing x=x(t)= the displacement of the first plow x(2)=0 Y=y(t)= the displacement of the second plow From the first problem we already knowA=5.1295 16. In this part of the problem let us do somegeneralization If = the time the first plow left the garageat distance a miles then since Andfor the second plow it is similar If= the time the first plow left the garageat distance b miles then since 17. And since they crash when Plugging in numbers from the problem or the second snow plow left at 2:25 p.m. 18. Dr. Huang has come up with a crash theoremfor this problem. One day it began to snow exactly at noon at aheavy and steady rate. Plow #1 left Garage #1 which is a miles fromthe intersection at ta p.m.; Plow #2 left Garage#2, which is b miles from the intersection at tbp.m. Then the two plows crash at theintersection of and only if Thus for this type of crashing problemdepends only of the difference a-b of thedistances. 19. If you remember from part (b) However in this problem we are onlyinterested in absolute distance from theintersection. Plugging in initial values we get. 20. Thereason we are only interested in theabsolute value of the distances is becausethe distance between the two plows at anytime t is given by Pythagorean Theoremwhich says the square of the distancesbetween the plows is and anynumber squared is always positive. So if we let Then 21. Now the minimum distance between the twoplows is when the derivative of the functionis =0 So 22. Now we know when they were the closest toeach other but remember they can only seeeach other if they get within .1 miles ofeach other. So Unfortunatelythe two plow drivers did not get the pleasure of seeing each others ugly mugs. 23. Also out of curiosity we can find how far each one was from the intersection.