From point a of the circular route whose length. From one point on the circular route

The same formulas are true: \[(\large(S=v\cdot t \quad \quad \quad v=\dfrac St \quad \quad \quad t=\dfrac Sv))\]
from one point in one direction with speeds \(v_1>v_2\) .

Then if \(l\) is the length of the circle, \(t_1\) is the time after which they will end up at the same point for the first time, then:

That is, for \(t_1\) the first body will go the distance\(l\) greater than the second body.

If \(t_n\) is the time after which they will end up at the same point for the \(n\) –th time, then the formula is valid: \[(\large(t_n=n\cdot t_1))\]

\(\blacktriangleright\) Let two bodies begin to move from different points in one direction with speeds \(v_1>v_2\) .

Then the problem easily reduces to the previous case: you first need to find the time \(t_1\) after which they will end up at the same point for the first time.
If at the moment of starting the movement the distance between them \(\buildrel\smile\over(A_1A_2)=s\), That:

Task 1 #2677

Task level: Easier than the Unified State Exam

Two athletes start in the same direction from diametrically opposite points on the circular track. They run with different constant speeds. It is known that at the moment when the athletes first caught up, they stopped training. How many more laps did the athlete run at a higher average speed than the other athlete?

Let's call the athlete with the higher average speed first. First, the first athlete had to run half a circle to reach the starting point of the second athlete. After that, he had to run as much as the second athlete ran (roughly speaking, after the first athlete ran half a circle, before the meeting he had to run every meter of the track that the second athlete ran, and the same number of times as the second athlete ran this meter ).

Thus, the first athlete ran \(0.5\) more laps.

Answer: 0.5

Task 2 #2115

Task level: Easier than the Unified State Exam

The cat Murzik runs in a circle from the dog Sharik. The speeds of Murzik and Sharik are constant. It is known that Murzik runs \(1.5\) times faster than Sharik and in \(10\) minutes they run two laps in total. How many minutes will it take Sharik to run one lap?

Since Murzik runs \(1.5\) times faster than Sharik, then in \(10\) minutes Murzik and Sharik in total run the same distance that Sharik would run in \(10\cdot (1 + 1.5) ) = 25\) minutes. Consequently, Sharik runs two circles in \(25\) minutes, then Sharik runs one circle in \(12.5\) minutes

Answer: 12.5

Task 3 #823

Task level: Equal to the Unified State Exam

From point A of a circular orbit distant planet Two meteorites flew out at the same time in the same direction. The speed of the first meteorite is 10,000 km/h greater than the speed of the second. It is known that for the first time after departure they met 8 hours later. Find the length of the orbit in kilometers.

At the moment when they first met, the difference in the distances they flew was equal to the length of the orbit.

In 8 hours the difference became \(8 \cdot 10000 = 80000\) km.

Answer: 80000

Task 4 #821

Task level: Equal to the Unified State Exam

A thief who stole a handbag runs away from the owner of the handbag along a circular road. The speed of the thief is 0.5 km/h greater than the speed of the owner of the handbag, who is running after him. In how many hours will the thief catch up with the owner of the handbag for the second time, if the length of the road along which they are running is 300 meters (assume that he caught up with her the first time after the theft of the handbag)?

First way:

The thief will catch up with the owner of the handbag for the second time at the moment when the distance he will run becomes 600 meters greater than the distance that the owner of the handbag will run (from the moment of theft).

Since his speed is \(0.5\) km/h higher, then in an hour he runs 500 meters more, then in \(1: 5 = 0.2\) hours he runs \(500: 5 = 100\) meters more. He will run 600 meters more in \(1 + 0.2 = 1.2\) hours.

Second way:

Let \(v\) km/h be the speed of the owner of the handbag, then
\(v + 0.5\) km/h – the speed of the thief.
Let \(t\) h be the time after which the thief will catch up with the owner of the handbag for the second time, then
\(v\cdot t\) – the distance that the owner of the handbag will run in \(t\) hours,
\((v + 0.5)\cdot t\) – the distance that the thief will cover in \(t\) hours.
The thief will catch up with the owner of the handbag for the second time at the moment when he runs exactly 2 laps more than her (that is, \(600\) m = \(0.6\) km), then \[(v + 0.5)\cdot t - v\cdot t = 0.6\qquad\Leftrightarrow\qquad 0.5\cdot t = 0.6,\] whence \(t = 1.2\) h.

Answer: 1.2

Task 5 #822

Task level: Equal to the Unified State Exam

Two motorcyclists start simultaneously from one point on a circular track in different directions. The speed of the first motorcyclist is twice that of the second. An hour after the start, they met for the third time (consider that the first time they met after the start). Find the speed of the first motorcyclist if the length of the road is 40 km. Give your answer in km/h.

At the moment when the motorcyclists met for the third time, the total distance they traveled was \(3 \cdot 40 = 120\) km.

Since the speed of the first is 2 times greater than the speed of the second, then out of 120 km he traveled a part 2 times greater than the second, that is, 80 km.

Since they met for the third time an hour later, the first one drove 80 km in an hour. Its speed is 80 km/h.

Answer: 80

Task 6 #824

Task level: Equal to the Unified State Exam

Two runners start simultaneously in the same direction from two diametrically opposite points on a circular track 400 meters long. How many minutes will it take for the runners to catch up for the first time if the first runner runs 1 kilometer more in an hour than the second?

In an hour, the first runner runs 1000 meters more than the second, which means he will run 100 meters more in \(60: 10 = 6\) minutes.

The initial distance between runners is 200 meters. They will be equal when the first runner runs 200 meters more than the second.

This will happen in \(2 \cdot 6 = 12\) minutes.

Answer: 12

Task 7 #825

Task level: Equal to the Unified State Exam

A tourist left city M along a circular road 220 kilometers long, and 55 minutes later a motorist followed him from city M. 5 minutes after departure he caught up with the tourist for the first time, and another 4 hours after that he caught up with him for the second time. Find the speed of the tourist. Give your answer in km/h.

First way:

After the first meeting, the motorist caught up with the tourist (for the second time) 4 hours later. By the time of the second meeting, the motorist had driven a circle more than the tourist had covered (that is, \(220\) km).

Since during these 4 hours the motorist overtook the tourist by \(220\) km, the speed of the motorist is \(220: 4 = 55\) km/h greater than the speed of the tourist.

Let now the speed of the tourist be \(v\) km/h, then he managed to walk before the first meeting \ the motorist managed to pass \[(v + 55)\dfrac(5)(60) = \dfrac(v + 55)(12)\ \text(km).\] Then \[\dfrac(v + 55)(12) = v,\] from where we find \(v = 5\) km/h.

Second way:

Let \(v\) km/h be the speed of the tourist.
Let \(w\) km/h be the speed of the motorist. Since \(55\) minutes \(+ 5\) minutes \(= 1\) hour, then
\(v\cdot 1\) km is the distance that the tourist traveled before the first meeting. Since \(5\) minutes \(= \dfrac(1)(12)\) hours, then
\(w\cdot \dfrac(1)(12)\) km – the distance that the motorist traveled before the first meeting. The distances they traveled before their first meeting are: \ Over the next 4 hours, the motorist drove more than the tourist covered in a circle (by \(220\) \ \

When using quantities in the exercise that are related to distance (speed, circle length), they can be solved by reducing them to movement in a straight line.

The greatest difficulty for schoolchildren in Moscow and other cities, as practice shows, is caused by tasks on Roundabout Circulation in the Unified State Examination, the search for an answer in which is associated with the use of an angle. To solve the exercise, the circumference can be specified as a part of a circle.

Repeat these and others algebraic formulas you can in the “Theoretical information” section. In order to learn how to apply them in practice, solve exercises on this topic in the “Catalogue”.

Lesson type: repeating and generalizing lesson.

Lesson objectives:

educational

various types

word problems

developing

educational

educational process

Lesson equipment:

interactive whiteboard;
envelopes with assignments, thematic control cards, consultant cards.

Lesson structure.

	Main stages of the lesson	Tasks to be solved at this stage
	Organizational moment, introductory part	Creation friendly atmosphere in class set students up for productive work identify absentees check students' readiness for the lesson
	Preparing students for active work(repetition)	test students’ knowledge on the topic: “Solving word problems of various types on movement” implementation of the development of speech and thinking of responding students development of analytical and critical thinking of students through commenting on classmates’ answers organize educational activities the whole class during the response of students called to the board
	Stage of generalization and systematization of the studied material (work in groups)	test students' ability to solve problems of various types types of movement, to form knowledge among students, reflected in the form of ideas and theories, the transition from particular ideas to broader generalizations carry out the formation of moral relations of students towards participants in the educational process (during group work)
	Examination doing the work, adjustment (if necessary)	check the execution of data for groups of tasks (their correctness) continue to develop in students the ability to analyze, highlight the main thing, build analogies, generalize and systematize develop discussion skills
	Summing up the lesson. Analysis homework	inform students about homework, explain how to complete it motivate the need and obligation to do homework summarize the lesson

Forms of organization cognitive activity students:

frontal form of cognitive activity - at stages II, IY, Y.
group form of cognitive activity - at stage III.

Teaching methods: verbal, visual, practical, explanatory - illustrative, reproductive, partially - search, analytical, comparative, generalizing, traductive.

Lesson progress

I. Organizational moment, introductory part.

The teacher announces the topic of the lesson, the objectives of the lesson and the main points of the lesson. Checks the class's readiness for work.

II. Preparing students for active work (repetition)

Answer the questions.

What kind of motion is called uniform (movement at a constant speed).
What is the formula for the path with uniform motion ( S = Vt).
From this formula, express the speed and time.
Specify units of measurement.

Conversion of speed units

III. Stage of generalization and systematization of the studied material (work in groups)

The whole class is divided into groups (5-6 people per group). It is advisable to have students in the same group different levels preparation. Among them, a group leader (the strongest student) is appointed, who will lead the work of the group.

All groups receive envelopes with assignments (they are the same for all groups), consultant cards (for weak students) and thematic control sheets. In the thematic control sheets, the group leader gives grades to each student in the group for each task and notes the difficulties that the students encountered when completing specific tasks.

Card with tasks for each group.

№ 5.

№ 7. Powerboat walked upstream 112 km and returned to the point of departure, spending 6 hours less on the return journey. Find the speed of the current if the speed of the boat in still water is 11 km/h. Give your answer in km/h.

No. 8. The motor ship travels along the river to its destination 513 km and, after stopping, returns to the point of departure. Find the speed of the ship in still water if the current speed is 4 km/h, the stay lasts 8 hours, and the ship returns to the point of departure 54 hours after departure. Give your answer in km/h.

No. 9. From pier A to pier B, the distance between which is 168 km, the first motor ship set off at a constant speed, and 2 hours after that, the second one set off after it, at a speed 2 km/h higher. Find the speed of the first ship if both ships arrived at point B at the same time. Give your answer in km/h.

Sample thematic control card.

Class ________ Student's full name___________________________________

Job No.		Comment

Cards consultants.

Card No. 1 (consultant)

1. Driving on a straight road

When solving problems involving uniform motion, two situations often occur.

If the initial distance between objects is S, and the velocities of the objects are V1 and V2, then:

a) when objects move towards each other, the time after which they will meet is equal to .

b) when objects move in one direction side time, through which the first object will catch up with the second, is equal to , ( V 2 > V 1)

Example 1. The train, having traveled 450 km, was stopped due to snow drift. Half an hour later the path was cleared, and the driver, increasing the speed of the train by 15 km/h, brought it to the station without delay. Find the initial speed of the train if the distance traveled by it to the stop was 75% of the total distance.

Let's find the entire path: 450: 0.75 = 600 (km)
Let's find the length of the second section: 600 – 450 =150 (km)
Let's create and solve the equation:

X= -75 does not fit the conditions of the problem, where x > 0.

Answer: The initial speed of the train is 60 km/h.

Card No. 2 (consultant)

2. Driving on a closed road

If the length of a closed road is S, and the speeds of objects V 1 and V 2, then:

a) when objects move in different directions, the time between their meetings is calculated by the formula;
b) when objects move in one direction, the time between their meetings is calculated by the formula

Example 2. At a competition on a circuit track, one skier completes a lap 2 minutes faster than the other and an hour later beats him by exactly a lap. How long does it take each skier to complete the circle?

Let S m – length of the ring route and x m/min and y m/min – speeds of the first and second skiers, respectively ( x> y) .

Then S/x min and S/y min – the time it takes the first and second skiers to complete the lap, respectively. From the first condition we obtain the equation. Since the speed of removal of the first skier from the second skier is ( x- y) m/min, then from the second condition we have the equation .

Let's solve the system of equations.

Let's make a replacement S/x= a And S/y= b, then the system of equations will take the form:

. Multiply both sides of the equation by 60 a(a + 2) > 0.

60(a + 2) – 60a = a(a + 2)a 2 + 2a- 120 = 0. Quadratic equation has one positive root a = 10 then b = 12. This means that the first skier completes the circle in 10 minutes, and the second skier in 12 minutes.

Answer: 10 min; 12 min.

Card No. 3 (consultant)

3. Movement along the river

If an object moves with the flow of a river, then its speed is equal to Vflow. =Vob. + Vcurrent

If an object moves against the flow of a river, then its speed is equal to Vagainst the current = V inc. – Vcurrent. The object’s own speed (speed in still water) is equal to

The speed of the river flow is

The speed of the raft is equal to the speed of the river flow.

Example 3. The boat went 50 km downstream of the river, and then traveled 36 km in the opposite direction, which took it 30 minutes longer than along the river. What is the boat's own speed if the speed of the river is 4 km/h?

Let the boat's own speed be X km/h, then its speed along the river is ( x+ 4) km/h, and against the flow of the river ( x- 4) km/h. The time it takes for the boat to move along the river flow is hours, and against the river flow is hours. Since 30 minutes = 1/2 hour, then according to the conditions of the problem we will create the equation =. Multiply both sides of the equation by 2( x+ 4)(x- 4) >0 .

We get 72( x+ 4) -100(x- 4) = (x+ 4)(x- 4) x 2 + 28x- 704 = 0 x 1 =16, x 2 = - 44 (excluded since x> 0).

So, the boat's own speed is 16 km/h.

Answer: 16 km/h.

IV. Problem solving analysis stage.

Problems that caused difficulty for students are analyzed.

No. 1. From two cities, the distance between which is 480 km, two cars simultaneously drove towards each other. After how many hours will the cars meet if their speeds are 75 km/h and 85 km/h?

75 + 85 = 160 (km/h) – approach speed.
480: 160 = 3 (h).

Answer: the cars will meet in 3 hours.

No. 2. From cities A and B, the distance between which is 330 km, two cars simultaneously left towards each other and met 3 hours later at a distance of 180 km from city B. Find the speed of the car leaving city A. Give the answer in km/ h.

(330 – 180) : 3 = 50 (km/h)

Answer: the speed of a car leaving city A is 50 km/h.

No. 3. A motorist and a cyclist left at the same time from point A to point B, the distance between which is 50 km. It is known that a motorist travels 65 km more per hour than a cyclist. Determine the speed of the cyclist if it is known that he arrived at point B 4 hours 20 minutes later than the motorist. Give your answer in km/h.

Let's make a table.

Let's create an equation, taking into account that 4 hours 20 minutes =

Obviously, x = -75 does not fit the conditions of the problem.

Answer: The cyclist's speed is 10 km/h.

No. 4. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 14 km. How many minutes will it take for the motorcyclists to meet each other for the first time if the speed of one of them is 21 km/h greater than the speed of the other?

Let's make a table.

Let's create an equation.

, where 1/3 hour = 20 minutes.

Answer: in 20 minutes the motorcyclists will pass each other for the first time.

No. 5. From one point on a circular track, the length of which is 12 km, two cars started simultaneously in the same direction. The speed of the first car is 101 km/h, and 20 minutes after the start it was one lap ahead of the second car. Find the speed of the second car. Give your answer in km/h.

Let's make a table.

Let's create an equation.

Answer: the speed of the second car is 65 km/h.

No. 6. A cyclist left point A of the circular track, and 40 minutes later a motorcyclist followed him. 8 minutes after departure, he caught up with the cyclist for the first time, and another 36 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 30 km. Give your answer in km/h.

Let's make a table.

	Movement before the first meeting

cyclist

1. Two cars left point A to point B at the same time. The first one drove at a constant speed all the way. The second car traveled the first half of the journey at a speed lower than the speed of the first by 15 km/h, and the second half of the journey at a speed of 90 km/h, as a result of which it arrived at point B at the same time as the first car. Find the speed of the first car if it is known to be greater than 54 km/h. Give your answer in km/h.

2. A train, moving uniformly at a speed of 60 km/h, passes a forest belt, the length of which is 400 meters, in 1 minute. Find the length of the train in meters.

3. The distance between cities A and B is 435 km. The first car drove from city A to city B at a speed of 60 km/h, and an hour after that the second car drove towards it at a speed of 65 km/h. At what distance from city A will the cars meet? Give your answer in kilometers.

4. By two parallel railway tracks A freight and a passenger train travel in the same direction at speeds of 40 km/h and 100 km/h, respectively. The length of a freight train is 750 m. Find the length of a passenger train if the time it takes to pass the freight train is 1 minute.

5. A train, moving uniformly at a speed of 63 km/h, passes a pedestrian walking in the same direction parallel to the tracks at a speed of 3 km/h in 57 seconds. Find the length of the train in meters.

6. Solving motion problems.

7. The road between points A and B consists of ascent and descent, and its length is 8 km. The pedestrian walked from A to B in 2 hours 45 minutes. The time it took to descend was 1 hour and 15 minutes. At what speed did the pedestrian walk downhill if his speed on the uphill is 2 km/h less than the speed on the downhill? Express your answer in km/h.

8. The car traveled from the city to the village in 3 hours. If he increased his speed by 25 km/h, he would spend 1 hour less on this journey. How many kilometers is the distance from the city to the village?

http://youtu.be/x64JkS0XcrU

9. Ski competitions take place on a circular track. The first skier completes one lap 2 minutes faster than the second and an hour later is exactly one lap ahead of the second. How many minutes does it take the second skier to complete one lap?

10. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 6 km. How many minutes will it take for the motorcyclists to meet each other for the first time if the speed of one of them is 18 km/h greater than the speed of the other?

Movement problems from Anna Denisova. Website http://easy-physics.ru/

11. Video lecture. 11 movement problems.

1. A cyclist travels 500 m less every minute than a motorcyclist, so he spends 2 hours more on a journey of 120 km. Find the speeds of the cyclist and motorcyclist.

2. A motorcyclist stopped to refuel for 12 minutes. After that, increasing the speed by 15 km/h, he caught up wasted time at a distance of 60 km. How fast was he moving after stopping?

3. Two motorcyclists set off simultaneously towards each other from points A and B, the distance between which is 600 km. While the first one covers 250 km, the second one manages to cover 200 km. Find the speed of motorcyclists if the first one arrives in B three hours earlier than the second one in A.

4. The plane was flying at a speed of 220 km/h. When he had 385 km less to fly than he had already covered, the plane increased its speed to 330 km/h. The average speed of the aircraft along the entire route turned out to be 250 km/h. How far did the plane fly before its speed increased?

5. By railway the distance from A to B is 88 km, by water it increases to 108 km. The train from A leaves 1 hour later than the ship and arrives at B 15 minutes earlier. Find the average speed of the train, if it is known that it is 40 km/h greater than the average speed of the ship.

6. Two cyclists left two places 270 km apart and are traveling towards each other. The second travels 1.5 km less per hour than the first, and meets him after as many hours as the first travels in kilometers per hour. Determine the speed of each cyclist.

7. Two trains depart from points A and B towards each other. If the train from A leaves two hours earlier than the train from B, then they will meet halfway. If they leave at the same time, then after two hours the distance between them will be 0.25 of the distance between points A and B. How many hours does each train take to complete the entire journey?

8. The train passed a man standing motionless on the platform in 6 s, and past a platform 150 m long in 15 s. Find the speed of the train and its length.

9. Train 1 km long passed the pole in 1 minute, and through the tunnel (from the entrance of the locomotive to the exit of the last car) at the same speed - in 3 minutes. What is the length of the tunnel (in km)?

10. Freight and fast trains departed simultaneously from stations A and B, the distance between which is 75 km, and met half an hour later. The freight train arrived at B 25 minutes later than the fast train at A. What is the speed of each train?

11. Piers A and B are located on a river whose current speed in this section is 4 km/h. A boat travels from A to B and back without stopping at an average speed of 6 km/h. Find the boat's own speed.

12. Video lecture. 8 problems for moving in a circle

12. Two points move uniformly and in the same direction along a circle 60 m long. One of them does full turn 5 seconds faster than the other. In this case, the points coincide every time after 1 minute. Find the speeds of the points.

13. How much time passes between two consecutive coincidences of the hour and minute hands on a watch dial?

14. Two runners start from one point on the stadium ring track, and the third - from a diametrically opposite point at the same time as them in the same direction. After running three laps, the third runner caught up with the second. Two and a half minutes later, the first runner caught up with the third. How many laps per minute does the second runner run if the first one overtakes him once every 6 minutes?

15. Three racers start simultaneously from one point on a track shaped like a circle and ride in the same direction at constant speeds. The first rider overtook the second for the first time, making his fifth lap, at a point diametrically opposite to the start, and half an hour after that he overtook the third rider for the second time, not counting the start. The second rider caught up with the third for the first time three hours after the start. How many laps per hour does the first driver make if the second driver completes the lap in at least 20 minutes?

16. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 14 km. How many minutes will it take for the motorcyclists to meet each other for the first time if the speed of one of them is 21 km/h greater than the speed of the other?

17. A cyclist left point A of the circular route, and 30 minutes later a motorcyclist followed him. After 10 min. after leaving, he caught up with the cyclist for the first time, and 30 minutes later he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 30 km. Give your answer in km/h.

18. A clock with hands shows 3 o’clock exactly. In how many minutes minute hand will it be aligned with the clock for the ninth time?

18.1 Two riders race. They will have to drive 60 laps along a 3 km long ring track. Both riders started at the same time, and the first one reached the finish line 10 minutes earlier than the second one. What was it equal to? average speed the second driver, if it is known that the first driver overtook the second driver for the first time after 15 minutes?

13. Video lecture. 6 problems for moving on water.

19. Cities A and B are located on the banks of a river, with city B downstream. At 9 o'clock in the morning a raft leaves from city A to city B. At the same moment, a boat departs from B to A and meets the raft 5 hours later. Having reached city A, the boat turns back and sails to B at the same time as the raft. Will the boat and raft arrive in city B by nine o'clock in the evening of the same day?

20. A motor boat left from point A to point B against the flow of the river. On the way, the engine broke down, and while it took 20 minutes to repair it, the boat was carried down the river. Determine how much later the boat arrived at point B, if the journey from A to B usually takes one and a half times longer than from B to A?

21. Cities A and B are located on the banks of a river, with city A downstream. From these cities two boats leave at the same time towards each other and meet in the middle between the cities. After the meeting, the boats continue their journey, and, having reached cities A and B, respectively, turn around and meet again at a distance of 20 km from the place of the first meeting. If the boats initially swam against the current, then the boat that left A would catch up with the boat that left B, 150 km from B. Find the distance between the cities.

22. Two steamships, the speed of which is the same in still water, depart from two piers: the first from A downstream, the second from B upstream. Each ship stays at its destination for 45 minutes and returns back. If the steamships depart simultaneously from their starting points, then they meet at point K, which is twice closer to A than to B. If the first steamship departs from A 1 hour later than the second one departs from B, then on the way back the steamships meet 20 km from A. If the first steamboat leaves from A 30 minutes earlier than the second from B, then on the way back they meet 5 km above K. Find the speed of the river and the time it takes the second steamboat to reach from A to TO.

23. A raft set off from point A to point B, located downstream of the river. At the same time, a boat came out to meet him from point B. Having met the raft, the boat immediately turned and sailed back. How far from A to B will the raft travel by the time the boat returns to point B, if the speed of the boat in still water is four times the speed of the current?

24. Piers A and B are located on a river whose current speed in this section is 4 km/h. A boat travels from A to B and back at an average speed of 6 km/h. Find the boat's own speed.

The video lecture "Solving text problems on movement in a circle and water" discusses all types of problems on movement in a circle and water from Open bank Unified State Examination tasks in mathematics.

You can get acquainted with the content of the video lecture and watch its fragment.

Problems for moving in a circle:

1. Two motorcyclists start simultaneously in the same direction from two diametrically opposite points on a circular track, the length of which is 7 km. How many minutes will it take for the motorcyclists to meet each other for the first time if the speed of one of them is 5 km/h greater than the speed of the other?

2. A cyclist left point A of the circular route, and 20 minutes later a motorcyclist followed him. 5 minutes after departure he caught up with the cyclist for the first time, and another 46 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 46 km. Give your answer in km/h.

3. The clock with hands shows 6 hours 45 minutes. In how many minutes will the minute hand line up with the hour hand for the fifth time?

4. Two drivers are racing. They will have to drive 22 laps along a 3 km long ring track. Both riders started at the same time, and the first one reached the finish line 11 minutes earlier than the second one. What was the average speed of the second driver, if it is known that the first driver overtook the second driver for the first time after 10 minutes?

Tasks for moving on water:

5. The motor boat traveled 72 km upstream of the river and returned to the point of departure, spending 6 hours less on the return journey. Find the speed of the boat in still water if the current speed is 3 km/h. Give your answer in km/h.

6. The distance between piers A and B is 72 km. A raft set off from A to B along the river, and 3 hours later a yacht set off after it, which, having arrived at point B, immediately turned back and returned to A. By this time the raft had covered 39 km. Find the speed of the yacht in still water if the speed of the river is 3 km/h. Give your answer in km/h.

7. The boat travels the distance from pier M to pier N along the river in 6 hours. One day, not reaching 40 km from pier N, the boat turned back and returned to pier M, spending 9 hours on the entire journey. Find the speed of the boat in still water , if the current speed is 2 km/h.

8. From point A, a boat and a raft simultaneously sailed down the river. Having passed 40/3 km, the boat turned back and, having passed 28/3 km, met the raft. You need to find the boat's own speed if you know that the current speed is 4 km/h.

9. The motor boat sailed across the lake, and then went down the river flowing from the lake. Path along the lake at 15% less way along the river. The time it takes a boat to move on a lake is 2% longer than on a river. What percentage is the current speed less? own speed boats?

10. In the spring, a boat moves against the flow of the river 1 2/3 times slower than with the flow. In summer, the current becomes 1 km/h slower, so in summer the boat goes against the river current 1 1/2 times slower than with the current. Find the speed of the current in spring (in km/h).

Fragment of video lecture:

More than 80,000 real Unified State Exam problems 2019

You are not logged in to the system "". This does not interfere with viewing and solving tasks Open Bank of Unified State Examination Problems in Mathematics, but to participate in the user competition to solve these tasks.

Search result for Unified State Exam assignments in mathematics for the query:
« A bicycle left point A of the circular track» — 251 tasks found

Task B14 ()

(views: 605 , answers: 13 )

A cyclist left point A of the circular track, and 10 minutes later a motorcyclist followed him. 2 minutes after departure, he caught up with the cyclist for the first time, and 3 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 5 km. Give your answer in km/h.

Task B14 ()

(views: 624 , answers: 11 )

A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and another 10 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 10 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 ()

(views: 691 , answers: 11 )

A cyclist left point A of the circular track, and 10 minutes later a motorcyclist followed him. 5 minutes after departure he caught up with the cyclist for the first time, and another 15 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 10 km. Give your answer in km/h.

Answer: 60

Task B14 ()

(views: 612 , answers: 11 )

A cyclist left point A of the circular track, and 30 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and another 47 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 47 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 ()

(views: 608 , answers: 9 )

A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and another 19 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 19 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 ()

(views: 618 , answers: 9 )

A cyclist left point A of the circular track, and 20 minutes later a motorcyclist followed him. 2 minutes after departure, he caught up with the cyclist for the first time, and another 30 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 50 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 ()

(views: 610 , answers: 9 )

A cyclist left point A of the circular track, and 30 minutes later a motorcyclist followed him. 5 minutes after departure, he caught up with the cyclist for the first time, and another 26 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 39 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 ()

(views: 622 , answers: 9 )

A cyclist left point A of the circular track, and 50 minutes later a motorcyclist followed him. 5 minutes after departure he caught up with the cyclist for the first time, and another 12 minutes after that he caught up with him for the second time. Find the speed of the motorcyclist if the length of the route is 20 km. Give your answer in km/h.

The correct answer has not yet been determined

Task B14 (