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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.
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 $$s_{n}$$.
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
If two objects( or persons) A and B are moving with speeds $$s_a\ \&\ s_b$$ km/h then the relative velocity between them
A) when they are moving in the same direction is $$\left(s_a\ -\ s_b\right)$$ km/hr
B) when they are moving in the opposite direction is $$s_a\ +\ s_b$$ km/hr
CLOCKS:
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.
If the speed of the boat downstream is x km/hr and the speed of the boat upstream is y km/hr, then
Speed of boat in still water= $$\dfrac{x+y}{2}$$ km/hr
Speed of stream= $$\dfrac{x-y}{2}$$ km/hr
If the speed a person is x m/s, and the speed of the escalator is y m/s, then the relative speed is (x+y) m/s (if in same direction).
If they are moving opposite direction, then the relative speed is (x-y) m/s.
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
$$\frac{S_{1}}{S_{2}}$$=$$\sqrt{\frac{T_{2}}{T_{1}}}$$
Linear Tracks(Races)
If A and B both participate in an L metre race, and A beats B by x metres(say), then the time taken by A to complete L metres is the same as the time taken by B to complete (L-x) metres.
In that case, if speed of A is a m/s and B is b m/s then $$\dfrac{L}{a}=\dfrac{L-x}{b}$$
So, the ratio of their speeds is $$\dfrac{a}{b}=\dfrac{L}{L-x}$$
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
If the speed is increased by $$\frac{a}{n}$$, the time will reduce by $$\frac{a}{n+a}$$ and vice versa.
Or If the speed is decreased by $$\frac{a}{n}$$, the time will increase by $$\frac{a}{n-a}$$ and vice versa
Because the Product of speed and time is distance, which remains constant. Hence, if one of the speeds or times increases, the other has to decrease.
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)
When a person is on an escalator, he can move in either one of the two ways.
If they are moving with the escalator, there will be fewer steps to climb than the total (Same direction= less steps)
If they are moving against the escalator, there will be more steps to climb than the total (Opposite directions = more steps)
Let us suppose that an escalator has a total of L stairs , and it churns out E stairs/second(moves at this speed). We consider a person who can climb P stairs/second.
When the person is moving on the escalator, he will move certain steps on his own, while the escalator moves.
Let us assume he climbs D stairs on his own. The time taken to do this would be $$\frac{D}{P}$$ seconds. In this time, the escalator will churn out $$\frac{D}{P}\times E$$ stairs.
So, the total number of steps when the person is
(A) Moving With the Escalator = steps climbed oneself + steps produced by escalator OR $$L=D+\frac{D}{P}\times E$$
(B) Moving Against the Escalator = steps climbed oneself – steps produced by escalator OR $$L=D-\frac{D}{P}\times E$$
