Constant Distance:
Let the distances travelled in each part of the journey be $$d_{1}, d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the speeds in each part be $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.
If $$d_{1} = d_{2} = d_{3} =...= d_{n}$$= d, then the average speed is the harmonic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.
Constant Time:
Let the distances travelled in each part of the journey be $$d_{1}, d_{2}, d_{3}$$ and so on till $$d_{n}$$ and the time taken for each part be $$t_{1}, t_{2}, t_{3}$$ and so on till $$t_{n}$$.
If $$t_{1} = t_{2} = t_{3} =...= t_{n}$$= t, then the average speed is the arithmetic mean of the speeds $$s_{1}, s_{2}, s_{3}$$ and so on till $$s_{n}$$.
The questions in this section can vary from being very easy to surprisingly difficult. This is a conceptual section (especially questions involving clocks) and some of the questions can consume a lot of time. While solving, write down the equations as far as possible to avoid mistakes. The few extra seconds can help you avoid silly mistakes. Also, check if the units of distance, speed and time match up. So if you see yourself adding a unit of distance like m to a unit of speed m/s, you would realize you have missed a term. Choose to apply the concept of relative speed wherever possible as it can greatly reduce the complexity of the problem. Like speed and distance, in time and work while working with terms ensure that you convert all terms to consistent units like man-hours.
CLOCKS:
PIPES & CISTERNS:
Inlet Pipe: A pipe which is used to fill the tank is known as Inlet Pipe.
Outlet Pipe: A pipe which can empty the tank is known as an Outlet Pipe.
The part filled per hour =$$\frac{1}{x}-\frac{1}{y}$$(if y>x)
The part emptied per hour =$$\frac{1}{y}-\frac{1}{x}$$(if x>y)
BOATS & STREAMS
If the speed of water is 'W' and speed of a boat in still water is 'B'
The direction along the stream is called downstream. And, the direction against the stream is called upstream.
Speed of boat in still water= $$\frac{x+y}{2}$$km/hr
Rate of stream= $$\frac{x-y}{2}$$km/hr
1m/s = 3.6 km/h
Circular Tracks
If two people are running on a circular track with speeds in the ratio a:b where a and b are co-prime, then
If two people are running on a circular track having perimeter I, with speeds m and n,
(when they are running in opposite directions)
(when they are running in the same direction)
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,
t = $$\sqrt{x*y}$$
where x = time taken (after the meeting) by P to reach B and y = time taken (after the meeting) by Q to reach A.
A and B started st a time towards each other. After crossing each other, they took $$T_{1}$$ hrs, $$T_{2}$$ hrs respectively to reach their destinations. If they travel at constant speeds $$S_{1}$$ and $$S_{2}$$ respectively all over the journey, Then
$$\frac{S_{1}}{S_{2}}$$=$$\sqrt{\frac{T_{2}}{T_{1}}}$$
Distance = Speed$$\times$$Time
Speed = $$\frac{Distance}{Time}$$
Time = $$\frac{Distance}{Speed}$$
While covering the Speed in m/s to km/hr. multiply it by 3.6. It is because 1m/s = 3.6 km/hr
If the ratio of the speeds of A and B is a : b, then
Average speed= $$\frac{Total Distance Travelled}{Total Time Taken}$$
If a part of a journey is travelled at speed $$S_{1}$$ km/hr in $$T_{1}$$ hours and the remaining part at speed $$S_{2}$$ km/hr in $$T_{2}$$ hours then.
Total distance travelled= $$S_{1}T_{1}$$+$$S_{2}T_{2}$$ km
Average speed=$$\frac{S_{1}T_{1}+S_{2}T_{2}}{T_{1}+T_{2}}$$ km/hr
If $$D_{1}$$ km is travelled at speed of $$S_{1}$$ km/hr, and $$D_{2}$$ km is travelled at speed of $$S_{2}$$ km/hr then
Average Speed= $$\frac{D_{1}+D_{2}}{\frac{D_{1}}{S_{1}}+\frac{D_{2}}{S_{2}}}$$ km/hr
Then the average speed during the whole journey is $$\frac{2xy}{x+y}$$ km/hr
Then the average speed during the whole journey is $$\frac{x+y}{2}$$ km/hr
TRAINS:
Two trains of length $$L_1$$ and $$L_2$$ travelling at speeds of $$S_1$$ and $$S_2$$ cross each other in
$$\frac{L_1+L_2}{S_1+S_2}$$ if they are going in opposite directions
Work:
The number of days taken by both of them together is $$\frac{x*y}{x+y}$$
$$\frac{M_{1}H_{1}D_{1}}{W_{1}}$$=$$\frac{M_{2}H_{2}D_{2}}{W_{2}}$$