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Races and Circular Motion Concept — Complete Formula and Tricks

Races and circular motion is an important and frequently tested topic in quantitative aptitude. It is asked in competitive exams including CAT, SSC CGL, SSC CHSL, Bank PO, Bank Clerk, Railway RRB and CSAT. A strong understanding of races and circular motion concept, formulas and tricks is essential for scoring well in these exams. In this post we cover everything from linear races and the concept of beat by distance and beat by time, ratio of speeds in a race, circular motion in same direction and opposite direction, time for first meeting on track, time for first meeting at starting point, number of times two runners meet on a circular track and dead heat — all explained with clear formulas and solved examples.

📚 What You Will Learn in This Post

Linear Races — Beat by Distance and Beat by Time Concept

Ratio of Speeds in a Race — Formula and Solved Examples

Circular Motion — Same Direction and Opposite Direction

Time for First Meeting on Circular Track — Formula

Time for First Meeting at Starting Point — LCM Method

Number of Times Two Runners Meet on Track — Formula

Dead Heat — Definition and Concept

                                       Races & Circular Motion
Concept:-

Linear RacesBeat by DistanceBeat by TimettABA, BTime is constant∴ Speed ∝ DistanceA beat B by distance d then A cover total distance D in t time while B cover (D - d) distance in t timettA, BA, BDistance is constant∴ Speed ∝ A beat B by time t. then A reaches end point in timet then B reaches end point intime t which is (t + t)Example: In a 1 km. race A beats B by 100 m. Find ratio of speeds of A & BSol:-Example: In a 1 km. race A beats B by 10 sec. Find ratio of speeds of A & BSol:-ttABA, B100 m A BDistane 1000 900 10 : 9Speed 10 : 9 Answer A BTime t (t + 10) Speed (t + 10) : t Answer

Example: In a 1 km. race A beats B by 50 m or 10 sec. What time does A take to completer the race.
Sol:-
A beats B by 50 m or 10 sec. So B will cover 50 m in 10 sec.
Speed of B = \(\frac{{50}}{{10}}\) = 5 m/sec
time taken by B to run 1 Km. = \(\frac{{1000}}{5}\) = 200 sec.
∴ time taken by A to run 1 km. = 200 – 10
= 190 sec                 Answer

Circular Motion (Race)ConceptBAAB

● When will first tim they meet ?
● When will they meet first time at starting point.
● How many times they will meet on the track.

BAAB2 m/sec5 m/sec210 m210 m2 m/sec5 m/sec● time for first time meeting(i). Answer● time taken by A to reachstarting point(ii) ∴ time to meet first time at the starting point = LCM(42, 105) = 210 Answer● Number of times they will meeton track = write their speed in simplest formso that there is no factor commonin them then subtract them to getnumber of times they will meet.⟹ 5 - 2 = 3 AnswerAnswer●(i) time for first time meeting = ●(ii) time taken by A to reach starting point time take by B to reach starting point = = 105 sec ∴ time to meet first time at the starting point = LCM(42, 105) = 210 Answer●(iii) Number of times they will meeton track = write their speed in simplest formso that there is no factor commonin them then add them to getnumber of times they will meet.⟹ 5 + 2 = 7 AnswerAnswer= 30 secAnswer

Dead Heat:- A dead Heat is a situation when all participants reaches the finish point at the same instant of time.

❓ Frequently Asked Questions on Races and Circular Motion

Q1. What is a race in quantitative aptitude?

A race is a contest of speed between two or more participants over a fixed distance. In quantitative aptitude races involve comparing speeds of participants to find who wins, by how much distance or by how much time. The concept of racing in mathematics uses the relationship between speed, distance and time to solve problems asked in CAT, SSC CGL, Bank PO and Railway exams.

Q2. What does beat by distance mean in races?

When A beats B by distance d in a race of total distance D it means that when A completes the full distance D, B has only covered D minus d distance in the same time. Since time is constant speed is proportional to distance. So the ratio of speeds of A and B equals the ratio of distances covered which is D : (D – d).

Q3. What does beat by time mean in races?

When A beats B by time t in a race it means A finishes the race in time T while B takes time T plus t to finish the same distance. Since distance is constant speed is inversely proportional to time. So the ratio of speeds of A and B equals T plus t : T.

Q4. When do two runners meet first time on a circular track?

When two runners A and B run in the same direction on a circular track of length L with speeds Sa and Sb respectively they first meet after time = L divided by (Sa – Sb). When they run in opposite directions they first meet after time = L divided by (Sa + Sb). This is one of the most important formulas in circular motion problems.

Q5. When do two runners meet first time at the starting point?

Two runners A and B meet first time at the starting point after time equal to LCM of the individual times taken by each runner to complete one full round of the circular track. For example if A takes 42 seconds and B takes 105 seconds for one round then they first meet at starting point after LCM of 42 and 105 which is 210 seconds.

Q6. How many times do two runners meet on a circular track?

To find the number of times two runners meet on a circular track write their speeds in simplest form with no common factor. If they run in the same direction subtract the smaller speed from the larger speed to get the number of meetings. If they run in opposite directions add the two speeds to get the number of meetings.

Q7. What is dead heat in races?

A dead heat is a situation in a race where all participants reach the finish point at exactly the same instant of time. In a dead heat no participant wins and all are declared equal. This concept is occasionally asked in competitive exams as part of races and time speed distance problems.

Q8. Where can I practice races and circular motion questions?

After understanding the concept you can practice on our Races and Circular Motion Exercise page which contains solved practice questions. You can also check our Time Speed and Distance Concept and Ratio and Proportion Concept for related topics.

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