Distance = Speed$$\times$$Time
Speed = $$\frac{Distance}{Time}$$
Time = $$\frac{Distance}{Speed}$$
This is one of the most fundamental formulas and is used extensively throughout the topic.
Speed - Distance - Time
Very Important
Question 1
Two trains cross each other in 14 seconds when running in opposite directions along parallel tracks. The faster train is 160 m long and crosses a lamp post in 12 seconds. If the speed of the other train is 6 km/hr less than the faster one, its length, in m, is
Solution
Speed of the faster train = $$\frac{160}{12}=\frac{40}{3}\ $$ m/s
Speed of the slower train = $$\frac{40}{3}-\left(6\times\ \frac{5}{18}\right)=\frac{35}{3}$$ m/s
Sum of speeds (when the trains travel towards each other) = $$\frac{40}{3}+\frac{35}{3}=25$$ m/s
Let the slower train be $$x$$ metres long; then: $$\frac{160+x}{25}=14$$
On solving, $$x=190\ m$$
Question 2
Two trains A and B were moving in opposite directions, their speeds being in the ratio 5 : 3. The front end of A crossed the rear end of B 46 seconds after the front ends of the trains had crossed each other. It took another 69 seconds for the rear ends of the trains to cross each other. The ratio of length of train A to that of train B is
Solution
Considering the length of train A = La, length of train B = Lb.
The speed of train A be 5*x, speed of train B be 3*x.
From the information provided :
The front end of A crossed the rear end of B 46 seconds after the front ends of the trains had crossed each other.
In this case, train A traveled a distance equivalent to the length of train B which is Lb at a speed of 5*x+3*x = 8*x because both the trains are traveling in the opposite direction.
Hence (8*x)*(46) = Lb.
In the information provided :
It took another 69 seconds for the rear ends of the trains to cross each other.
In the next 69 seconds
The train B traveled a distance equivalent to the length of train A in this 69 seconds.
Hence (8*x)*(69) = La.
La/Lb = 69/46 = 3/2 = 3 : 2
Question 3
Mira and Amal walk along a circular track, starting from the same point at the same time. If they walk in the same direction, then in 45 minutes, Amal completes exactly 3 more rounds than Mira. If they walk in opposite directions, then they meet for the first time exactly after 3 minutes. The number of rounds Mira walks in one hour is
Solution
Correct Answer: 8
Considering the distance travelled by Mira in one minute = M,
The distance traveled by Amal in one minute = A.
Given if they walk in the opposite direction it takes 3 minutes for both of them to meet. Hence 3*(A+M) = C. (1)
C is the circumference of the circle.
Similarly, it is mentioned that if both of them walk in the same direction Amal completes 3 more rounds than Mira :
Hence 45*(A-M) = 3C. (2)
Multiplying (1)*15 we have :
45A + 45M = 15C.
45A - 45M = 3C.
Adding the two we have A = $$\frac{18C}{90}$$
Subtracting the two M = $$\frac{12C}{90}$$
Since Mira travels $$\frac{12C}{90}$$ in one minute, in one hour she travels :$$\frac{12C}{90}\cdot60\ =\ 8C$$
Hence a total of 8 rounds.
Alternatively,
Let the length of track be L
and velocity of Mira be a and Amal be b
Now when they meet after 45 minutes Amal completes 3 more rounds than Mira
so we can say they met for the 3rd time moving in the same direction
so we can say they met for the first time after 15 minutes
So we know Time to meet = Relative distance /Relative velocity
so we get $$\frac{15}{60}=\frac{L}{a-b}$$ (1)
Now When they move in opposite direction
They meet after 3 minutes
so we get $$\frac{3}{60}=\frac{L}{a+b}$$ (2)
Dividing (1) and (2)
we get $$\frac{\left(a+b\right)}{\left(a-b\right)}=5$$
or 4a =6b
or a = 3b/2
Now substituting in (1)
we get :
$$\frac{L}{b}\times\ 2=\ \frac{15}{60}$$
so $$\frac{L}{b}\ =\frac{1}{8}$$
So we can say 1 round is covered in $$\frac{1}{8}$$ hours
so in 1-hour total rounds covered = 8.
Question 4
Trains A and B start traveling at the same time towards each other with constant speeds from stations X and Y, respectively. Train A reaches station Y in 10 minutes while train B takes 9 minutes to reach station X after meeting train A. Then the total time taken, in minutes, by train B to travel from station Y to station X is
Solution
M - First meeting point
Let the speeds of trains A and B be 'a' and 'b', respectively.
$$\dfrac{x}{a}=\ \dfrac{\ D-x}{b}$$
It is given,
$$\dfrac{D}{a}=10$$ and $$\dfrac{x}{b}=9$$
$$\dfrac{x}{\dfrac{D}{10}}=\ \dfrac{\ D-x}{\dfrac{x}{9}}$$
$$\dfrac{10x}{D}=\ \dfrac{\ 9D-9x}{x}$$
$$10x^2=\ \ 9D^2-9Dx$$
$$10x^2+9Dx-9D^2=\ 0$$
Solving, we get $$x=\dfrac{3D}{5}$$
$$\dfrac{x}{b}=9$$
$$\dfrac{3D}{b\times5}=9$$
$$\dfrac{D}{b}=15$$
The total time taken by train B to travel from station Y to station X is 15 minutes.
The answer is option B
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