TIME , SPEED & DISTANCE
The proportionalities Implicit in The Equation S × T = D
The above equation has three implicit proportionality dimensions each of which has its own critical bearing on the solving of time, speed and distance problems.
1. Direct proportionality between time and distance (when the speed is constant) time µ distance
2. Direct Proportionality between speed and distance (when the time is constant)
speed µ distance
Relative movement, therefore, can be viewed as the movement of one body relative to another moving body.
Case I : Two bodies are moving in opposite directions at speeds S1 and S2 respectively.
The relative sped is defined as S1 + S2
Case II : Two bodies are moving in the same direction.
The relative speed is defined as
(a) S1 – S2 when S1 is greater than S2.
(b) S2 – S1 when S1 is lesser than S2.
Motion is a Straight Line
A. Two or more bodies starting from the same point and moving in the same direction: Their relative speed is S1 – S2
B. Moving in the opposite direction: Their relative speed will be initially given by S1 + S2.
The basic relation for trains problems is the same: Speed × Time = Distance.
(a) When the train is crossing a moving object, the speed has to be taken as the relative speed of the train with respect to the object. All the rules for relative speed will apply for calculating the relative speed.
(b) The distance to be covered when crossing an object whenever a train crosses an object will be equal to; Length of train + Length of object
ST = Speed of train S0 = Speed of object t = time
LT = Length of train S0 = Length of object
Ex. 1 At what time between 2 – 3 p.m. is the first right angle in that time formed by the hands of the clock ?
Sol. At 2 p.m. the minute hand can be visualised as being 10 kilometres behind the hour hand. (considering the clock dial to be
a race track of circumference 60 km such that each minute represents a kilometre).
Also, the first right angle between 2 – 3 is formed when the minute hand is 15 kilometres ahead of teh hour hand. Thus, the
minute hand has to cover 25 kilometres over the hour hand.
This can be written using the unitary method:
Distance covered by the minute hand over the hour hand.
55 kilometres __________________ in 1 hour
25 kilometres ___________________ in what time?
® 5/11 of an hour.
Thus, the first right angle between 2 – 3 is formed at 5/11 hours past 2 o’clock.
This can be converted into minutes and seconds using unitary method again as :
1 hour ___________________ 60 minutes
5/11 hours ________________ ? minutes
® 300/11 minutes = 27 (3/11) minutes
1 minute ________________ 60 seconds
3/11 minutes ____________ ? seconds ® 180/11
seconds = 16.3636 seconds.
Hence, the required answer is : 2 : 27 : 16.36 seconds.
ASSIGNMENT
1. The sinhagad express left pune at noon sharp. Two hours later, the Deccan Queen started from Pune in the same direction.
The Deccan Queen overtook the sinhagad express at 8 p.m. find the average speed of the two trains over this journey if the sum of their average speeds is 70 km/h.
(A) 34.28 km/h (B) 35 km/h
(C) 50 km/h (D) 12 km/h
(E) 16 km/h
2. Walking at 3/4 of his normal speed, abhishek is 16 minutes late in reaching his office. The usual time taken by him to cover the distance between his home and his office is
(A) 48 minutes (B) 60 minutes (C) 42 minutes (D) 62 minutes
(D) 66 minutes
3. Ram and Bharat travel the same distance at the rate of 6 km per hour and 10 km per hour respectively. If Ram takes 30 minutes longer than Bharat, the distance travelled by each is
(A) 67 km (B) 10 km
(C) 7.5 km (D) 20 km
(E) 8 km/h
4. Two trains for Mumbai leave Delhi at 6 a.m. and 6 : 45 am and travel at 100 kmph and 136 kmph respectively. How many kilometres from Delhi will the two trains be together
(A) 262.4 km (B) 260 km
(C) 283.33 km (D) 275 km
(E) None of these
5. Two trains, Calcutta Mail and Bombay Mail, start at the same time from stations Calcutta and Bombay respectively towards each other. After passing
each other, they take 12 hours and 3 hours to reach Bombay and Calcutta respetively. If the Calcutta mail is moving at the sped of 48 km/h, the speed of the bombay mail is
(A) 24 km/h (B) 22 km/h
(C) 21 km/h (D) 96 km/h
(E) 120 km/h
6. A cyclist moving on a circular track of radius 100 metres completes one revolution in 2 minutes. What is the average speed of cyclist (approximately) ?
(A) 314 m/minute (B) 200 m/minute (C) 300 m/minute (D) 900 m/minute
(E) None of these
7. A person travelled a distance of 200 kilometre between two cities by a car covering the first quarter of the journey at a constant speed of
40 km/h and the remaining three quarters at a constant speed of x km/h. If the average speed of the person for the entire journey was 53.33 km/h what is the value of x.
(A) 55 km/h (B) 60 km/h
(C) 70 km/h (D) 80 km/h
(E) 100 km/h
8. A car travels 1/3 of the distance on a straight road with a velocity of 10 km/h, the next 1/3 with a velocity of 20 km/h and the last 1/3
with a velocity of 60 km/h. What is the average velocity of the car for the whole journey ?
(A) 18 km/h (B) 10 km/h
(C) 20 km/h (D) 15 km/h
(E) None of these
9. Two cars started simultaneously toward each other from town A and B, that are 480 km apart. It took the first car travelling from A to
B 8 hours to cover the distance and the second car travelling from B to A 12 hours. Determine at what distance from A the two cars meet.
(A) 288 km (B) 200 km
(C) 300 km (D) 196 km
(E) None of these
10. Walking at 3/4 of his usual speed, a man is 16 minutes late for his office. The uausl time taken by him to cover that distance is
(A) 48 minutes (B) 60 minutes (C) 42 minutes (D) 62 minutes
(E) 75 minutes
11. Two people started simultaneously towards each other from Siliguri and Darjeeling, which are 60 km apart. They met 5 hours later
. After their meeting, the first person, who travelled from Siliguri to Darjeeling, decreased his speed by 1.5 km/h and the other person,
who travelled from Darjeeling to Siliguri, increased his speed by 1.5 km/h. The first person is known to arrive at Darjeeling 2.5
hours earlier than the second person arrived at Siliguri. Find the initial speed of the first person.
(A) 4.5 km/h (B) 6 km/h
(C) 7.5 km/h (D) 9 km/h
(E) 8 km/h
12. Two friends Arun and Nishit, on their last day in colege, decided to meet after 20 years on a river. Arun had to sail 42 km to the
meeting place and nishit had to sail 35 5/7 per cent less. To arrive at the meeting place at the same time as his friend Nishit, Arun started
at the same time as Nishit and sailed with speed exceeing by 5 km/h the speed of Nishit. Find the speed of Arun.
(A) 10 kmph (B) 14 km/h
(C) 9 kmph (D) Both b and c
(E) None of these
13. A motorboat moves from point A to point B and back again, both points being located on the river bank. If the speed of the boat
in still water is doubled, then the trip from A to B and back again would take 20% of the time that the motorboat usually
spends in the journey. How many times is the actual speed of the launch higher than the speed of the river flow ?
ANSWER KEY
1) A 2) A 3) C 4) C 5) D 6) A 7) B 8) A 9) A 10) A 11 ) C 12) B 13) A