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Motion in a Straight Line

This is one of the primary areas of application of time, speed and distance. By using the basic relationship between time, speed and distance the following three different cases are possible:

Case 1
 When S (Distance) is constant
V α Description: 3135.png
 
So, V1/V2 = T2/T1
 
It is read as V is inversely proportional to T.
 
It can be understood in terms of the reciprocal-ratio-multiplication relationship between V and T.
 
Suppose distance = 1000 km and Speed = 100 km/h
 
Speed = 100 km/h Time = 10 h
 
Speed = 100 × 2 Time = 1/2 × 10
 
Speed = 100 × 3 Time = 1/3 × 10
 
Speed = 100 × 1/2 Time = 2 × 10
 
To simplify it, the product stability ratio can be further used. So, the more the speed, the lesser is the time taken and the lesser the speed, more will be the time taken.
 
Example-1
A man cycles with a speed of 10 km/h and reaches his office at 1 pm. However, when he cycles with a speed of 15 km/h, he reaches his office at 11 am At what speed should he cycle so that he reaches his office at 12 noon?
Solution
Using the product stability ratio,
 
The speed is being increased by 50%, so the time taken will reduce by 33.33%.
 
So, 33.33 % of Time = 2 h
 
Hence, total time = 6 h
 
So, distance = 10 × 6 = 60 km
 
This distance is to be covered in 5 h
 
So, speed = 60/5 = 12 km/h
 
Alternatively, it can be seen that time taken in three (Given) situations are in AP. Hence, speeds will be in HP. Required Speed = Harmonic Mean of two speeds.
 
So, Required Speed = Description: 9289.png = 12 km/h
 
 
Example-2
Siddharth goes by a bike to pick up his girlfriend everyday from college and then drops her at her house. College timings are till 5 pm daily, but today the college at 4 pm His girlfriend, not finding Siddharth at the college gate, starts walking towards her house. Siddharth, unaware of this fact, leaves his house as usual meets his girlfriend on way, picks her up and drops her at her house. At night, Siddharth realizes that he had saved 40 minutes that day. What is the ratio of the speed of Siddharth to that of his girlfriend? (Both of them live in the same building)
Solution
Let us see the following schematic representation:
 

 
The usual route of Siddharth is home-college-home His route today is—home—meeting point—home.
 
And, in this way, 40 minutes are saved. So, he takes 20 minutes to cover the distance between the meeting point and the college. It can be further concluded that he usually reaches college at 5 pm, but today he reached at 4:40 (20 minutes are saved) and his girlfriend took 40 minutes. (she starts at 4 pm) to cover the distance between her college to the meeting point.
 
The ratio of time of Siddharth and his girlfriend = 20:40 = 1:2
 
The ratio of the speed of Siddharth and his girlfriend = 2:1
 

Case 2 When T (Time) is constant
 
S α V
 
So, S1/S2 = V1/V2
 
The higher is the speed, the more will be the distance covered and the lower the speed, the lesser will be the distance covered.

We will see that T constant is a situation specific to meeting point cases.
 
Example-3
Distance between two points AB = 110 km. Manoj starts running from A at a speed of 60 km/h and Ravi starts running from B at a speed of 40 km/h at the same time. They meet at a point X, somewhere on the line AB. What is ratio of AX to BX?
A X B
Solution
Since both Manoj and Ravi are running for the same time, T is constant. Hence, the ratio of the distance covered by them will be the same as the ratio of their speed.
 
So, AX/BX = Speed of Manoj/Speed of Ravi = 60/40 = 3:2
 
 
Some typical meeting point cases
When two persons are running between the ends of a linear track for infinite time:
 
Example-4
Two robots Mango and Bango start from the opposite ends A and B of a linear track respectively and keep running between the ends for infinite time. They meet for the first time at a point 60 m from A. If AB = 100 m, which point is their point of 4th meeting?
Solution
1st meeting point
 
The ratio of the speed of Mango and Bango is 60:40 = 3:2. Now Mango is moving towards B and Bango is moving towards A.
 
For the 2nd meeting, Bango reaches A after covering 60 m, Mango must have covered 90 m in the same time. So, he is at the mid-point of A and B. Now, the distance between Mango and Bango is 50 m. They will cover this distance of 50 m in the ratio of 3:2.
 
So, the point of their 2nd meeting is
 
 
It can be seen here that the sum of the distance covered by both Mango and Bango from the starting till now is 300 m, and the difference between the distance covered between the 1st meeting and the 2nd meeting is 200 m.
 
So, again they will meet for the 3rd time when they have covered a total distance of 200 m together. Mango and Bango will cover this distance of 200 m in the ratio of 3:2.
 
Distance covered by Mango = 120 m and distance covered by Bango = 80 m.
 
So, the 3rd meeting point is point B.
 
To have a 4th meeting, they will again have to cover a total distance of 200 m. So, the point of their 4th meeting is 20 m from A.
 
So, we can now generalize the above situation as:
 
Distance covered by both of them for the 1st meeting = 100 m
 
Distance covered by both of them for the 2nd meeting = 100 m + 200 m = 300 m
 
Distance covered by both of them for the 3rd meeting = 300 m + 200 m = 500 m
 
Distance covered by both of them for the 4th meeting = 500 m + 200 m = 700 m
 
The ratio of the speed of Mango and Bango is 3:2, so the distance covered by Mango = 420 m
 
Hence, they will meet at 20 m from A.
 
 
Example-5
Two persons, Ram and Mohan, start from the same end A of a linear track AB and keep running to and fro for infinite time. They meet for the first time at a point 20 m from B. If AB = 100 m, which point is their point of 4th meeting?
Solution
Using the above generalization, distance covered by both of them for the 1st meeting = 200 m
 
Distance covered by both of them for the 2nd meeting = 200 m + 200 m = 400 m
 
Distance covered by both of them for the 3rd meeting = 400 m + 200 m = 600 m
 
Distance covered by both of them for the 4th meeting = 600 m + 200 m = 800 m
 
The ratio of the speed of Ram and Mohan is 3:2, so the distance covered by Ram = 480 m
 
Hence, they will meet at 80 m from A.
 
 

Limitation of above generalization For the meeting to occur after every 200 m, the ratio of the speed of the two runners should be less than 2. If it is more than or equal to 2, then the problems can only be evaluated on the basis of actual calculation.
 
Case 3 When V (Speed) is constant
 
S α T
 
So, S1/S2 = T1/T2
 
In layman terms, if a person is running with a speed of 20 km/h, then the ratio of the distance covered in one hour to the distance covered in two hours will be 1:2.




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