Speed, Time and Distance Concepts for CAT

TSD's basic ideas are used to solve problems involving motion in a straight line, relative motion, circle motion, trains, boats, clocks, races, and so on. Aside from that, these ideas can also be used to make new kinds of problems. Your ability to figure out how to solve these problems will only rest on how well you understand them.

Typically, 4-6 problems are asked on CAT exam every year. In addition to the CAT, nearly every aptitude test includes queries on TSD concepts.

---

Understanding Motion

[Click Here for Previous Year Questions]

When a body goes from point A to point B at a distance of D, it takes some time (T) and a certain speed (S) to cross that distance.

Here's how T, S, and D are connected: T × S = D i.e., Time × Speed = Distance

Time = \(\frac{Distance}{Speed}\)

Average Speed = \(\frac{Toatal \ Distance}{Total\ Time\ Taken}\)

When Distances are equal:

Average speed= \(\frac{2xy}{x+y}\)         (x & y are speeds)

Average speed= \(\frac{3xyz}{xy+yz+zx}\)     (x, y & z are speeds)

If t₁ and t₂ be the original and changed time and S₁ and S₂ be the original and changed speeds, then

S₁ x S₂ = \(\frac{Distance\ * (S_1\sim S_2)}{(t_1 \sim t_2) }\)

t₁x t₂ = \(\frac{Distance\ * (t_1\sim t_2)}{(S_1 \sim S_2) }\)

---

Relative Motion with Two or More Bodies

[Click Here for Previous Year Questions]

• When two bodies move in the same direction: If the speeds of the two bodies A and B be S_A and S_B, then their relative speed = S_A −S_B or S_B −S_A i.e., in the same direction the relative speed or effective speed between two bodies is the difference of their speeds. (The difference is always considered as positive)

• When two bodies move in the opposite direction: If the speeds of the two bodies A and B be S_A andS_B, then their relative speed = S _A + S_B .
  i. e. , in the opposite direction the relative speed or effective speed between two bodies is the sum of their speeds.

• Remember If speed of A is S_A and speed of B is S_B and A takes t_A time to cover MQ and B takes t_B time to cover

MP, then \(\frac{S_A}{S_B}\) = \(\sqrt\frac{t_B}{t_A}\)

---

To and Fro Motion in a Straight Line

[Click Here for Previous Year Questions]

_{This idea is merely an extension of the notion of relative motion between multiple dynamic (or moving) bodies.}

(a) When two bodies start moving towards each other

If the initial distance (or gap) between two bodies A and B is D, and their respective velocities are S_A and S_B, then A and B must cover D units of distance to meet for the first time, regardless of their respective speeds.

• To meet, they travel distances proportional to their individual velocities.
• If the initial distance (or separation) between two bodies A and B is D, then A and B must travel D units of distance to meet for the first time.
• For each successive meeting (e.g., second, third, fourth, etc.), both A and B must travel an additional 2D distance; for example, to meet for the fourth time, they must travel a total of D + (3 x 2D) = 7D units of distance. For the seventh meeting, they must travel D + (6 x 2D) = 13D units of distance together. Therefore, for each successive meeting, they must travel an additional 2D distance.

NOTE:

Individually, they will cover the distances in proportion to their respective velocities regardless of the number of meetings. Thus, the total distance travelled for the nth meeting is equal to (2n – 1)D.

• When two bodies start moving towards the same direction

If the two bodies A and B begin at the same end of the track, their respective velocities are S_A and S_B, and the length of the track is D, then A and B must cover 2D units of distance to meet for the first time, regardless of their individual speeds.

• If the distance between two specific points (or locations) is D unit, they must traverse 2D distance for their initial meeting after beginning to travel. Since the quicker body reaches the next (or opposite) end before the slower body, and since the faster body begins its return before the slower body reaches the same opposite end, the two bodies meet somewhere between the two ends after travelling their respective distances at their individual velocities.
• For each consecutive meeting, they must travel 2D units further than the previous meeting. Therefore, for the nth rendezvous, they must traverse (n 2D) units of distance together.
• At any instant, the distances travelled by the bodies are proportional to the ratio of their velocities.

---

Concept Based on Trains

[Click Here for Previous Year Questions]

• If two trains (or people) are going in different directions, their relative speed will be the same as the sum of their individual speeds.
• When two trains are going in the same direction, the difference between their speeds will be their relative speed.
• The distance between two points is always the same as the sum of their lengths.
• The distance to be travelled, such as a bridge, station, etc., is always equal to the sum of the length of the train and the length of the object.
• The distance between a pole, a person, a tree, etc., and the train is always equal to the length of the train.
• If a man is riding a train, the distance he has to go to cross another train is equal to the length of the train that is going or crossing him. In this case, the speeds of both trains will be taken into account.

---

Concept Based on Boats and Rivers (or streams)

[Click Here for Previous Year Questions]

• When the boat and the river's stream (or current) move in the same direction, the boat's relative speed is the sum of the speeds of the boat and the river. This is what is called "downstream speed."
• When the boat goes against the flow of the river, or in the opposite way, its speed is the difference between its speed and the speed of the stream. It's called "upstream speed."

Let B be the speed of the boat in still water and C be the speed of the river current.

If B>C;

Downstream Speed D = (B + C)

Upstream Speed U = (B – C)

Speed of the boat in still water/ Speed of river B = \(\frac{D+ U}{2}\)

Speed of current (or stream) C = \(\frac{D-U}{2}\)

When the distance a boat travels upstream (against the flow of water) is the same as the distance it travels downstream (with the flow of water), then,

 \(\frac{Time \ Taken \ by \ Boat \ in\ DS}{Time \ Taken\ by\ Boat\ in \ US}\) = \(\frac{Upstream \ Speed}{Downstream \ Speed}\)

---

Races

[Click Here for Previous Year Questions]

Terminology

(i) Startup or headstart - When a runner lets another runner stay ahead of them in the same race, this is called a startup in the race.

(ii) Dead heat - When two or more runners cross the finish line at the same time, they are considered to have "finished" the race in a "dead heat."

Some More Useful Concepts

• When it is said that A can give B a start of x metres in a race of y meters, it means that in the same amount of time, B runs x metres less than A.
• In a race of y meters, if A beats B by t seconds, B is the loser and A is the winner. When A crosses the finish line, B is still a ways behind A. So, it takes B t seconds to go the rest of the way. So, we can figure out how fast loser (B) is going.
• Throughout the race there is always a certain relationship among runners i.e., they always maintain the ratio of speeds.

---

Circular Motion

[Click Here for Previous Year Questions]

• When the entities are moving in opposite directions, their relative velocity equals the sum of their individual velocities.
• When two objects are travelling in the same direction, the speed between them is equal to the difference in their absolute speeds.

First Meeting

Let A and B be two runners

Time taken by them to meet for the first time = \(\frac{Length \ of \ Circular\ Track}{Relative\ Speed}\)

If A is the quickest runner and A meets B after t_AB hours, A meets C after t_AC hours, A meets D after t_AD seconds/hours, etc., then A has met all the other runners for the first time. The LCM of the times it takes for A, B, C, D, etc. to finally meet for the first time is t_AB.

First Meeting at the Starting Point

Assume that runner A is the quickest of all the runners and that he or she will meet runner B for the first time in t_AB hours, runner C for the first time in t_AC hours, runner D for the first time in t_AD seconds/hours, and so on. The LCM of t_AB, t_AC, t_AD, etc. is the time it takes for all of them to meet for the first time.