Speed, distance, and time is one of the most popular and important topics in the mathematics or quants section of any competitive exam. The concept of speed, time, and distance is used extensively for questions relating to different topics such as motion in a straight line, circular motion, boats and streams, races, clocks, etc.
Relationship between speed, time, and distance
- Speed = Distance/Time
This tells us how slow or fast an object moves. It describes the distance travelled divided by the time taken to cover the distance.
- Speed is directly proportional to distance and inversely proportional to time.
- Hence, Distance = Speed × Time
- Time = Distance/Speed,
- As speed increases, the time taken will decrease and vice versa.
- While converting speed from m/s to km/h, multiply it by 18/5.
- While converting speed from km/h to m/s, multiply it by 5/18.
If the ratio of the speeds of A and B is a:b, then
- The ratio of the times taken to cover the same distance is 1/a:1/b or b: a.
- The ratio of distance travelled in equal time intervals is a:b.
Average speed
- When the distance is constant, average speed = 2xy/(x + y); where x and y are the two speeds at which the same distance has been covered.
- When the time taken is constant, average speed = (x + y)/2; where, x and y are the two speeds at which we travelled for the same time.
- If the distance covered is constant in each part of the journey, then the average speed is the harmonic mean of the values.
- If the time taken is constant in each part of the journey, then the average speed is the arithmetic mean of the values.
Circular tracks
If two people are running on a circular track with speeds in ratio a:b, where a and b are co-prime, then
- They will meet at (a + b) distinct points if they are running in opposite directions.
- They will meet at |a - b| distinct points if they are running in the same direction.
If two people are running on a circular track having perimeter P, with speeds a and b,
- The time of their first meeting = P/(a + b) (when they are running in opposite directions)
- The time for their first meeting = P/(|a - b|) (when they are running in the same direction)
Some important points
If a person P starts from A and heads towards B and another person Q starts from B and heads towards A and they meet after a time 't' then,
Where, x = time taken (after meeting) by P to reach B and y = time taken (after meeting) by Q to reach A
- A and B start moving towards each other. After crossing each other, they took T1 hours and T2 hours, respectively, to reach their destinations. If they travel at constant speeds S1 and S2 throughout the journey, then
Boats and streams
Let the speed of water be 'W' and the speed of a boat in still water is 'B‘.
- The speed of the boat going downstream is B+W.
- The speed of the boat going upstream is B – W.
- The direction along the stream is called downstream.
- The direction against the stream is called upstream.
- If the speed of boat downstream is x km/h and the speed of the boat upstream is y km/h, then
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