1. INTRODUCTION PROBLEMS ON KINEMATICS

1 ON KINEMATICSJaan KaldaTranslation partially by Taavi PungasVersion: 29th November 20171 INTRODUCTIONFor a majority of physics PROBLEMS , solving can be reduced tousing a relatively small number of ideas (this also applies toother disciplines, mathematics). In order to become goodat problem solving, one must learn these ideas. However, it isnot enough if you onlyknowthe ideas: you also need to learnhow torecognizewhich ideas are to be used for a given prob-lem With experience it becomes clear that usually problemsactually contain hints about which ideas need to be text attempts to summarise the main ideas en-countered in solving KINEMATICS PROBLEMS (though, some ofthese ideas are more universal, and can be applied to someproblems of other elds of physics).

2 For each idea, there areone or several illustrative PROBLEMS . First you should try tosolve the PROBLEMS while keeping in mind those ideas whichare suggested for the given problem. If this turns out to betoo difficult, you can look at the hints for each problem,rather detailed hints are given in the respective section. It isintentional that there are no full solutions: just reading thesolutions and agreeing to what is written is not the best wayof polishing your problem solving skills. However, there is asection of answers you can check if your results are are also revision PROBLEMS for which there are no sugges-tions provided in the text: it is your task to gure out whichideas can be used (there are still hints).

3 PROBLEMS are classi ed as beingsimple,normal, anddifficult(the problem numbers are coloured according to thiscolour code). Please keep in mind that difficulty levels are relat-ive and individual categories: some problem marked as difficultmay be simple for you, and vice versa. As a rule of thumb, aproblem has been classi ed as a simple one if it makes use ofonly one idea (unless it is a really tricky idea), and a difficultone if the solution involves three or more is assumed that the reader is familiar with the conceptsof speed, velocity and acceleration, radian as the measure forangles, angular speed and angular acceleration, trigonometricfunctions and quadratic equations. In few places, derivativesand differentials are used, so a basic understanding of these con-cepts is also advisable (however, one can skip the appropriatesections during the rst reading).

4 2 VELOCITIES idea 1:Choose the most appropriate frame of reference. Youcan choose several ones, and switch between them as useful frames are where: some bodies are at rest; some projections of velocities vanish; motion is is recommended to investigate process in all potentially use-ful frames of reference. As mentioned above, in a good frameof reference, some velocity or its component (or accelerationor its component) vanishes or two velocities are equal. Oncea suitable frame of reference has been found, we may changeback into the laboratory frame and transform the now knownvelocities-accelerations using the rule of adding velocities (ac-celerations). NB! the accelerations can be added in the sameway as velocitiesonly ifthe frame's motion is translational ( does not rotate).

5 Pr a river coast, there is a port; when a barge passedthe port, a motor boat departed from the port to a village atthe distances1= 15 kmdownstream. It reached its destinationaftert= 45 min, turned around, and started immediately mov-ing back towards the starting point. At the distances2= 9 kmfrom the village, it met the barge. What is the speed of theriver water, and what is the speed of the boat with respect tothe water? Note that the barge did not move with respect tothe , the motion takes place relative to the water, whichgives us a hint: let us try solving the problem when using thewater frame of reference. If we look at things closer, it be-comes clear that this is, indeed, a good choice: in that frame,the speed of the boat is constant, and barge is at rest, themotion of the bodies is much simpler than in the coastal frameof planes y at the same height with speedsv1=800 km=handv2= 600 km=h, respectively.

6 The planes ap-proach each other; at a certain moment of time, the plane tra-jectories are perpendicular to each other and both planes are atthe distancea= 20 kmfrom the intersection points of their tra-jectories. Find the minimal distance between the planes duringtheir ight assuming their velocities will remain idea1advises us that we should look for a frame wheresome bodies are at rest; that would be the frame of one of theplanes. However, here we have a two-dimensional motion, sothe velocities need to be added and subtracted 1:A scalar quantity is a quantity which can be fullydescribed by a single numerical value only; a vector quantity isa quantity which needs to be described by a magnitude (alsoreferred to as modulus or length), and a direction.

7 The sumof two vectors aand bis de ned so that if the vectors are in-terpreted as displacements (the modulus of a vector gives thedistance, and its direction the direction of the displacement)then the vector a+ bcorresponds to the net displacement asa result of two sequentially performed displacements aand b. page 1 corresponds to the triangle rule of addition, see is de ned as the reverse operation of addition: if a+ b= cthen a= c b.+ a~a~b~a~ b~b~b~a~After having been introduced the concept of vectors, we canalso x our 2:Velocityis a vectorial quantity which can be de nedby the projections to the axes v= (vx;vy;vz);speedis themodulus of a vector,v=j vj= v2x+v2y+v2z. Similarly,dis-placementis a vector pointing from the starting point of a bodyto its nal position; travelleddistanceis the sum of the moduliof all the elementary displacements (the curve length).

8 For vectorial addition, there are two options. First, we canselect two axes, for instancexandy, and work with the respect-ive projections of the velocity vectors. So, if our frame moveswith the velocity uand the velocity of a body in that frame is vthen its velocity in the lab frame is w= u+ v, which can befound via projectionswx=vx+uxandwy=vy+uy. Altern-atively, we can approach geometrically and apply the trianglerule of addition, see we have chosen the reference frame of one of theplanes, the problem2can be solved by using the 2:For PROBLEMS involving addition of vectors (velocities,forces), the PROBLEMS can be often reduced to the application ofsimple geometrical facts, such as (a) the shortest path from apoint to a line (or plane) is perpendicular to the line (plane); (b)among such trianglesABCwhich have two xed side lengthsjBCj=aandjACj=b<a, the triangle of largest\ABChas\BAC= 90.

9 The next problem requires the application of several ideasand because of that, it is classi ed as a difficult problem. Whenswitching between reference frames, the following ideas will 3:Try to reveal hidden symmetries, and make the prob-lem into a symmetric 4:It is possible to gure out everything about a velocityor acceleration once we know one of its components and thedirection of the ' way of stating it is that a right-angled triangleis determined by one angle and one of its sides. For example,if we know that velocity is at angle to the horizontal and itshorizontal component iswthen its modulus isw=sin .pr of two rings with radiusris at rest and the othermoves at velocityvtowards the rst one.

10 Find how the velocityof the upper point of intersection depends ona, the distancebetween two rings' idea4can be used again in the following problem:pr with constant ascending velocity can be usedto investigate wind velocities at various heights. The givengraph of elevation angle against time was obtained by observinga such balloon. The balloon was released at distanceL= 1 kmfrom the point of observation and it seemed to be rising dir-ectly upwards. Knowing that wind velocity near the groundwas zero, nd the balloon's height at timet= 7 minafter itsstart and wind velocity at this In order to answer to the rst question here, we need also thefollowing 5:If a graph ofyversusxis given, quite often some tan-gent line and its slopedydxturn out to be useful.