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Next Page �. If there's no mention of stream speed in the question, assume it to be the speed of boat in still water. So, water stream increases the speed of boat. Hence, speeds are added while going downstream. So, water stream decreases the speed of boat due to opposite flow. Hence, speeds are subtracted while going upstream.

If you just remember this concept, you won't need to remember and recall the above formula while solving the problems.

Halving this would give you the speed of the current. Once you know how to deal with these 4 types of problems, the chapter should be an easy one for you.

In complex problems, students tend to get confused regarding the usage of this equation and often end up mixing up the speed, time and distance of different motions of different bodies. It must be mentioned here that this formula is the cornerstone of the chapter Time, Speed and Distance. Besides, this formula is also the source of the various formulae applied to the problems on the applications of time, speed and distance�to trains, boats and streams, clocks and races, circular motion and straight line motion.

In the equation above, speed can be defined as the rate at which distance is covered during the motion. It is measured in terms of distance per unit time and may have any combination of units of distance and time in the numerator and the denominator respectively.

When we say that the speed of a body is S kmph, we mean to say that the body moves with S kmph towards or away from a stationary point as the case may be. The unit used for measuring time is synchronous with the denominator of the unit used for measuring speed.

The above equation, as is self-evident, is such that the interrelationship between the three parameters defines the value of the third parameter if two of the three are known. Hence we can safely say that if we know two of the three variables describing the motion, then the motion is fully described and every aspect of it is known. The above equation has three implicit proportionality dimensions each of which has its own critical bearing on the solving of time, speed and distance problems.

A car moves for 2 hours at a speed of 25 kmph and another car moves for 3 hours at the same speed. Find the ratio of distances covered by the two cars. Note: This can be verified by looking at the actual distances travelled�being 50 km and 75 km in this case.

Here two motions of one body are being described and between these two motions the time is constant hence speed will be proportional to the distance ravelled. They meet at a point C�. Here again, the speed is directly proportional to the distance since two motions are described where the time of both the motions is the same, that is, it is evident here that the first and the second car travel for the same time.

Find the ratio of the distances travelled at the two speeds. Solution: Since time is constant between the two motions described, we can use the proportionality between speed and distance. Alternatively, you can also think in terms of percentage as d 2 will be The distance between A and B is km.

They meet at a point 40 km from A. Find the ratio of their speeds. Solution: For the bodies to meet, the time of travel is constant since the two cars have moved simultaneously. Here two motions of one body are being described and between these two motions the distance travelled is constant. Hence he speed will be inversely proportional to the time travelled for. They meet at a point C and reach their respective destinations B and A in t 1 and t 2 hours respectively Here again, the speed is inversely proportional to the time since two motions are described where the distance of both the motions is the same, that is, it is evident here that the first and the second car travel for the distance, viz.

Due to this, it is 20 minutes late. Find the original time for the journey beyond the point of accident. Find the average speed. If his speed had been 7. Find the distance from his house to his office. Solution: Notice that here the distance is constant. Hence, speed is inversely proportional to time. Alternatively, using the Product Constancy Table from the chapter of percentages. Normally, when we talk about the movement of a body, we mean the movement of the body with respect to a stationary point.

Relative movement, therefore, can be viewed as the movement of one body relative to another moving body. The following formulae apply for the relative speed of two independent bodies with respect to each other:.

Case I: Two bodies are moving in opposite directions at speeds S 1 and S 2 respectively. In other words, the relative speed can also be defined as the positive value of the difference between the two speeds, that is, S 1 � S 2. Problems on situations of motion in a straight line are one of the most commonly asked questions in the CAT and other aptitude exams.

Hence a proper understanding of the following concepts and their application to problem solving will be extremely important for the student.

Two or more bodies starting from the same point and moving in the same direction: Their relative speed is S 1 � S 2. However, if both the bodies reverse their direction at the same instant, there will be no change in the relative speed equation. In this case, the description of the motion of the two bodies between two consecutive meetings will also be governed by the proportionality between speed and distance since the time of movement between any two meetings will be constant.

Distances covered in this case: For every meeting, he total distance covered by the two bodies will be 2 D where D is the distance between the extreme points. However, notice that the value of 2 D would be applicable only if both the bodies reverse the direction between two meetings. In case only one body has reversed direction, the total distance would need to be calculated on a case-by-case basis.

The respective coverage of the distance is in the ratio of the individual speeds. Here again, if the two bodies start simultaneously, their movement will be governed by the direct proportionality between speed and distance.

The faster body will reach its extreme point first followed by the slower body reaching its extreme point next. Relative speed will change every time; one of the bodies reverses direction. The position of the meeting point will be determined by the ratio of the speeds of the bodies since the 2 movements can be described as having the time constant between them.

Distances covered in the above case: For the first meeting, the total distance covered by the two bodies will be D the distance between the extreme points. The coverage of the distance is in the ratio of the individual speeds. Thereafter, as the bodies separate and start coming together, the combined distance to be covered is 2 D. Note that if only one body is reversing direction between two meetings, this would not be the case and you will have to work it out.

They meet at a point 0. Find the point of their fourth meeting. Also, total distance to be covered by the two together for the fourth meeting is 7 D. This distance is divided in a ratio of 3 : 2 and thus we have that A will cover 4. A, having moved a distance of 4.

This is the required answer. After half an hour, B starts from the same place and walks in the same direction as A at a uniform speed and overtakes A after 1 hour 48 minutes.

Calculate the speed of B. Solution: Start solving as you read the question. From the first two sentences you see that A is 1 km ahead of B when B starts moving.

Important: The student is advised to take a closer look and get a closer understanding of these concepts by taking a few examples with absolute values of speed, time and distance. Try to visualise how two bodies separate and then come together.

The base area of the beaker is 24 cm 2. The water level rises by 10 cm every second. How quickly will the water level rise in a beaker with a base area of 30 cm 2. Note: In case of confusion in such questions the student is advised to use dimensional analysis to understand what to multiply and what to divide. How long will it take for. At the rate of 1.

If you had to find out the average speed of the whole journey, what would you do? However, this situation can be solved very conveniently using the process of alligation as explained below:.

Since, the two speeds are known to us, we will also know their ratio. The ratio of times for the two parts of the journey will then be the inverse ratio of the ratio of speeds.

Since the distance for the two journeys are equal. The answer will be the weighted average of the two speeds weighted on the basis of the time travelled at each speed.

What is the average speed of the car for the entire journey. The process of alligation, will be used here to give the answer as Note: For the process of alligation, refer to the chapter of Alligations. Note here, that since the speed ratio is , the value of the time ratio used for calculating the weighted average will be For instance, if the car goes km at a speed of 66kmph and km at a speed of kmph, what will be the average speed?

In this case the speed ratio being i. This would have been the ratio to be used for the time ratio in case the distances were the same for both the speeds. But since the distances are different, we cannot use this ratio in this form. The problem is overcome by multiplying this ratio by the distance ratio in this case it is to get a value of This is the ratio which has to be applied for the respective weights.

Hence, the alligation will look like:. Thus the required answer is 90 kmph. The student is advised to check this value through normal mathematical processes. Trains are a special case in questions related to time, speed and distance because they have their own theory and distinct situations. All the rules for relative speed will apply for calculating the relative speed.

Thus, the following cases will yield separate equations, which will govern the crossing of the object by the train:. This can be visualised by remembering yourself stationary on a railway platform and being crossed by a train. You would say that the train starts crossing you when the engine of the train comes in line with you.

Thus, the train would have travelled its own length in crossing you]. If the length of the train is metres, find the speed of the train. Solution: In this case, it is evident that the situation is one of the train crossing a stationary object without length. Hence, Case I is applicable here. However, the train requires 25 seconds to cross the same man if the trains are travelling in the same direction. If the length of the first train is metres and that of the train in which the man is sitting is metres, find the speed of the first train.

Solution: Here, the student should understand that the situation is one of the train crossing a moving object without length. Then applying the relevant formulae after considering the directions of the movements we get the equations:. However, as in the case of trains the adjustments to be made for solving questions on boats and streams are:. The boat has a speed of its own, which is also called the speed of the boat in still water S B. Another variable that is used in boats and streams problems is the speed of the stream S S.

The time of movement and the distance to be covered are to be judged by the content of the problem. Circular motion: A special case of movement is when two or more bodies are moving around a circular track. The relative speed of two bodies moving around a circle in the same direction is taken as S 1 � S 2. The peculiarity inherent in moving around a circle in the same direction is that when the faster body overtakes the slower body it goes ahead of it.

And for every unit time that elapses, the faster body keeps increasing the distance by which the slower body is behind the faster body. However, when the distance by which the faster body is in front of the slower body becomes equal to the circumference of the circle around which the two bodies are moving, the faster body again comes in line with the slower body.

This event is called as overlapping or lapping of the slower body by the faster body. We say that the slower body has been lapped or overlapped by the faster body. First meeting: Three or more bodies start moving simultaneously from the same point on the circumference of the circle, in the same direction around the circle. They will first meet again in the LCM of the times that the fastest runner takes in totally overlapping each of the slower runners.

First meeting at starting point: Three or more bodies start moving simultaneously from the same point on the circumference of a circle, in the same direction around the circle. Their first meeting at the starting point will occur after a time that is got by the LCM of the times that each of the bodies takes to complete one full round.

Problems on clocks are based on the movement of the minute hand and that of the hour hand as well as on the relative movement between the two. In my opinion, it is best to solve problems on clocks by considering a clock to be a circular track having a circumference of 60 km and each kilometre being represented by one minute on the dial of the clock. Then, we can look at the minute hand as a runner running at the speed of 60 kmph while we can also look at the hour hand as a runner running at an average speed of 5 kmph.

Since, the minute hand and the hour hand are both moving in the same direction, the relative speed of the minute hand with respect to the hour hand is 55 kmph, that is, for every hour elapsed, the minute hand goes 55 km minute more than the hour hand. Beyond this slight adjustment, the problems of clocks require a good understanding of unitary method. This will be well illustrated through the solved example below. Number of right angles formed by a clock: A clock makes 2 right angles between any 2 hours.

Thus, for instance, there are 2 right angles formed between 12 to 1 or between 1 and 2 or between 2 and 3 or between 3 and 4 and so on. However, contrary to expectations, the clock does not make 48 right angles in a day. This happens because whenever the clock passes between the time period 2�4 or between the time period 8�10 there are not 4 but only 3 right angles. This happens because the second right angle between 2�3 or 8�9 and the first right angle between 3�4 or 9�10 are one and the same, occurring at 3 or 9.

Right angles are formed when the distance between the minute hand and the hour hand is equal to 15 minutes. Exactly the same situation holds true for the formation of straight lines. There are 2 straight lines in every hour. However, the second straight line between 5�6 or 11�12 and the first straight line between 6�7 or 12�1 coincide with each other and are represented by the straight line formed at 6 or Straight lines are formed when the distance between he minute hand and the hour hand is equal to either 0 minutes or 30 minutes.

At what time between 2�3 p. Solution: At 2 p. Also, the first right angle between 2�3 is formed when the minute hand is 15 kilometres ahead of the hour hand. The Sinhagad Express left Pune at noon sharp. Two hours later, the Deccan Queen started from Pune in the same direction.

The Deccan Queen overtook the Sinhagad Express at 8 p. The usual time taken by him to cover the distance between his home and his office is. Ram and Bharat travel the same distance at the rate of 6 km per hour and 10 km per hour respectively. If Ram takes 30 minutes longer than Bharat, the distance travelled by each is. Two trains for Mumbai leave Delhi at 6 : 00 a. How many kilometres from Delhi will the two trains be together? Two trains, Calcutta Mail and Bombay Mail, start at the same time from stations Kolkata and Mumbai respectively towards each other.

After passing each other, they take 12 hours and 3 hours to reach Mumbai and Kolkata respectively. The distance between any 2 of them is 4 km. Shyam starts walking from his gym in a direction parallel to the road connecting his office and his house and stops when he reaches a point directly east of his office. He then reverses direction and walks till he reaches a point directly south of his office.

The total distance walked by Shyam is. Lonavala and Khandala are two stations km apart. How far from Lonavala will they will cross each other? Find the normal time. Alok walks to a viewpoint and returns to the starting point by his car and thus takes a total time of 6 hours 45 minutes. He would have gained 2 hours by driving both ways. How long would it have taken for him to walk both ways? Sambhu beats Kalu by 30 metres or 10 seconds. How much time was taken by Sambhu to complete a race meters.

On an average, how many minutes per hour does the train stop during the journey? Rajdhani Express travels km in 5 h and another km in 10 h. What is the average speed of train? What is the average velocity? The average speed of the car is:. What is the distance of his college from his house? If she takes 10 hours in all, what is the distance between her office and her home? A motor car does a journey in Find the distance of the journey.

Sujit covers a distance in 40 minutes if he drives at a speed of 60 kilometer per hour on an average. The whole journey took 5 hours. What is the distance he covered on the car? A railway passenger counts the telegraph poles on the rail road as he passes them. The telegraph poles are at a distance of 50 metres. What will be his count in 4 hours, if the speed of the train is 45 km per hour? Two trains A and B start simultaneously in the opposite direction from two points A and B and arrive at their destinations 9 and 4 hours respectively after their meeting each other.

At what rate does the second train B travel if the first train travels at 80 km per hour. Vinay fires two bullets from the same place at an interval of 12 minutes but Raju sitting in a train approaching the place hears the second report 11 minutes 30 seconds after the first.

What is the approximate speed of train if sound travels at the speed of metre per second? The pedestrian could see the car for 6 minutes and it was visible to him up to a distance of 0. What was the speed of the car? A journey of km takes 2 hours less by a fast train than by a slow train. If the average speed of the slow train be 16 kmph less than that of fast train, what is the average speed of the faster train?

A passenger train takes 2 h less for a journey of kilometres if its speed is increased by 5 kmph over its usual speed. Find the usual speed. A plane left half an hour later than the scheduled time and in order to reach its destination kilometre away in time, it had to increase its speed by Find its increased speed.

What is the average speed of the train? A cyclist moving on a circular track of radius metres completes one revolution in 2 minutes. What is the average speed of cyclist approximately? If the average speed of the person for the entire journey was What is the average velocity of the car for the whole journey?

Two cars started simultaneously toward each other from town A and B , that are km apart. It took the first car travelling from A to B 8 hours to cover the distance and the second car travelling from B to A 12 hours.

Determine at what distance from A the two cars meet. The usual time taken by him to cover that distance is. A and B travel the same distance at the rate of 8 kilometre and 10 kilometre an hour respectively. If A takes 30 minutes longer than B , the distance travelled by B is.

Two trains for Patna leave Delhi at 6 a. How many kilometres from Delhi will the two trains meet? After passing each other, they take 12 hours and 3 hours to reach Y and X respectively.

X and Y are two stations km apart. How far from X they will cross each other? A starts from a point that is on the circumference of a circle, moves metre in the North direction and then again moves metre East and reaches a point diametrically opposite the starting point.

Find the diameter of the circle? Ram and Shyam run a race of m. First, Ram gives Shyam a start of m and beats him by 30 s. Next, Ram gives Shyam a start of 3 min and is beaten by metres. Find the time in minutes in which Ram and Shyam can run the race separately. A motorboat went downstream for 28 km and immediately returned. It took the boat twice as long to make the return trip. If the speed of the river flow were twice as high, the trip downstream and back would take minutes.

Find the speed of the boat in still water and the speed of the river flow. A train requires 7 seconds to pass a pole while it requires 25 seconds to cross a stationary train which is metres long. Find the speed of the train. A boat sails downstream from point A to point B , which is 10 km away from A , and then returns to A. What must the actual speed of the boat for the trip from A to B to take exactly minutes? A boat sails down the river for 10 km and then up the river for 6 km.

What should be the minimum speed of the boat for the trip to take a maximum of 4 hours? If the river is running at 3 km per hour, it takes him 45 minutes to row to a place and back.

How far is the place? A boat goes 40 km upstream in 8 h and a distance of 49 km downstream in 7 h. The speed of the boat in still water is. Two trains are running on parallel lines in the same direction at speeds of 40 kmph and 20 kmph respectively.

The faster train crosses a man in the second train in 36 seconds. The length of the faster train is. A distance of 8 km, going upstream, is covered in. A boat goes 15 km upstream in 80 minutes. A boat can go from A to B and back in 6 h 30 minutes while it goes from A to C in 9 h.

How long would it take to go from C to A? Two trains pass each other on parallel lines. Each train is metres long. When they are going in the same direction, the faster one takes 60 seconds to pass the other completely. If they are going in opposite directions they pass each other completely in 10 seconds. Vinay runs metres in 20 seconds and Ajay runs the same distance in 25 seconds. By what distance will Vinay beat Ajay in a hundred metre race?

In a m race, Shyam runs at 1. At a game of billiards, A can give B 15 points in 60 and A can give C 20 in How many points can B give C in a game of 90? In a m race, the ratio of speed of two runners Vinay and Shyam is 3 : 4. If Vinay has a start of m then Vinay wins by. The faster train crosses a man in the slower train in 18 seconds. Find the length of the faster train. Two trains for Howrah leave Muzaffarpur at a. How many kilometres from Muzaffarpur will the two trains meet?

How many minutes per hour does the train stop? A boat rows 16 km up the stream and 30 km downstream taking 5 h each time. The velocity of the current. Vijay can row a certain distance downstream in 6 h and return the same distance in 9 h. When the river is running at 1. How far in the place? A man can row 30 km upstream and 44 km downstream in 10 hours. It is also known that he can row 40 km upstream and 55 km downstream in 13 hours. Find the speed of the man in still water.

The length of the minutes hand of a clock is 8 cm. Find the distance travelled by its outer end in 15 minutes. Between 5 a. A motorboat went down the river for 14 km and then up the river for 9 km. It took a total of 5 hours for the entire journey. Calculate the speed of the river flow if the round trip took a total of 20 hours. In a race of meters, Ajay beats Vijay by 60 metres and in a race of meters Vijay beats Anjay by 25 meters.

By how many meters will Ajay beat Anjay in a meter race? A motorboat whose speed in still water is 15 kmph goes 30 km downstream and comes back in a total 4 hours 30 min. Determine the speed of the stream. Ayrton Senna had to cover a distance of 60 km. Find the speed at which he travelled during the journey described.

Amitabh covered a distance of 96 km two hours faster than he had planned to. This he achieved by travelling 1 km more every hour than he intended to cover every 1 hour 15 minutes.

What was the speed at which Amitabh travelled during the journey? An urgent message had to be delivered from the house of the Peshwas in Pune to Shivaji who was camping in Bangalore.

A horse rider travels on horse back from Pune to Bangalore at a constant speed. Find the distance between Pune and Bangalore. A pedestrian and a cyclist start simultaneously towards each other from Aurangabad and Paithan which are 40 km apart and meet 2 hours after the start. Then they resumed their trips and the cyclist arrives at Aurangabad 7 hours 30 minutes earlier than the pedestrian arrives at Paithan. Which of these could be the speed of the pedestrian?

Two motorists met at 10 a. After their meeting, one of them proceeded in the East direction while the other proceeded in the North direction. Exactly at noon, they were 60 km apart. Two cyclists start simultaneously towards each other from Aurangabad and Ellora, which are 28 km apart. An hour later they meet and keep pedalling with the same speed without stopping. The second cyclist arrives at Ellora 35 minutes later than the first arrives at Aurangabad.

Find the speed of the cyclist who started from Ellora. Two ants start simultaneously from two ant holes towards each other. A bus left point X for point Y.

Two hours later a car left point X for Y and arrived at Y at the same time as the bus. If the car and the bus left simultaneously from the opposite ends X and Y towards each other, they would meet 1.

How much time did it take the bus to travel from X to Y? A racetrack is in the form of a right triangle. The longer of the legs of the track is 2 km more than the shorter of the legs both these legs being on a highway. The start and end points are also connected o each other through a side road. Find the length of the racetrack.

Two planes move along a circle of circumference 1. When they move in different directions, they meet every 15 seconds and when they move in the same direction, one plane overtakes the other every 60 seconds. Find the speed of the slower plane. Karim, a tourist leaves Ellora on a bicycle. Having travelled for 1. What distance will they cover before Rahim overtakes Karim? A tourist covered a journey partly by foot and partly by tonga.

He walked for 90 km and rode the tonga for 10 km. He spent 4 h less on the tonga than on walking. If the tourist had reversed the times he travelled by foot and on tonga, the distances travelled on each part of the journey would be equal. How long did he ride the tonga? Two Indian tourists in the US cycled towards each other, one from point A and the other from point B.

The first tourist left point A 6 hrs later than the second left point B , and it turned out on their meeting that he had travelled 12 km less than the second tourist. After their meeting, they kept cycling with the same speed, and the first tourist arrived at B 8 hours later and the second arrived at A 9 hours later. Find the speed of the faster tourist. Two joggers left Delhi for Noida simultaneously. The first jogger stopped 42 min later when he was 1 km short of Noida and the other one stopped 52 min later when he was 2 km short of Noida.

If the first jogger jogged as many kilometres as the second and the second as many kilometres as the first, the first one would need 17 min less than the second. Find the distance between Delhi and Noida. A tank of m 3 capacity is full of water. As a result the pump needs 16 min less to discharge the fuel than to fill up the tank. Find the filling capacity of the pump. An ant climbing up a vertical pole ascends 12 meters and slips down 5 meters in every alternate hour.

If the pole is 63 meters high how long will it take it to reach the top? Two ports A and B are km apart. Two ships leave A for B such that the second leaves 8 hours after the first.

The ships arrive at B simultaneously. An ant moved for several seconds and covered 3 mm in the first second and 4 mm more in each successive second than in its predecessor. If the ant had covered 1 mm in the first second and 8 mm more in each successive second, then the difference between the path it would cover during the same time and the actual path would be more than 6 mm but less than 30 mm.

Find the time for which the ant moved in seconds. The Sabarmati Express left Ahmedabad for Mumbai. Having travelled km, which constitutes Half an hour later, the track was cleared and the engine-driver, having increased the speed by 15 km per hour, arrived at Mumbai on time. Find the initial speed of the Sabarmati Express. Two swimmers started simultaneously from the beach, one to the south and the other to the east.

Two hours later, the distance between them turned out to be km. A motorcyclist left point A for point B. Two hours later, another motorcyclist left A for B and arrived at B at the same time as the first motorcyclist. Had both the motorcyclists started simultaneously from A and B travelling towards each other, they would have met in 80 minutes.

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