MATHS - 4.TIME AND DISTANCE

Time and distance is one topic in which many school students raise questions. We discussed this aspect in a tricky problem earlier, but now let us highlight some important concepts of it.

Distance travelled ( d )= speed (s) * time(t)………..............I

Every one understands and uses this main formula.

The only difficulty, few students experience is in conversion of units .The one needed often in American schools is

1 mile/hr = 22/15 feet/sec….

while converting ,the best practice is to multiply with unit ratio every time putting the units every time and tallying in the end whether we got the correct units or not. Let us see
with an example.

We want to convert 60 m/h to f/s say.

First write the conversion factor. It is

1 m/h = 22/15 f/s…write it as unit ratio…that is..

[1 m/h]/[22/15 f/s]=1…..or……[22/15 f/s]/[1 m/h]=1

[15/22 (m*s/f*h)]=1……or……[22/15 (f*h/m*s]= 1

now first put the given data with units and multiply with the above unit ratio putting the units.

[60 m/h][22/15 (f*h/m*s)]…if we put the correct unit ratio, as we did here, the required units will only remain. Others get cancelled out. Here miles and hours cancel out leaving the required feet and seconds only.

So we get 88 f/s only remaining …..now just see what happens if we put the ratio in a wrong way…we get

[60 m/h][15/22 (m*s/f*h)] giving us

60*15/22 (m*m*s/f*h*h)….we do not get f/s as we got last time.this shows that this unit ratio used is not correct and we should use the reciprocal ratio.

RELATIVE SPEED:

Coming to the main part of it now, many do not understand the concept of relative speed.

Suppose A and B are traveling from same place say P at the same time with speeds of s1 and s2. m/h respectively.

Case1.They are traveling in opposite directions.

So in 1 hr A will go s1 miles towards north say

In the same 1 hr B will go s2 miles towards south.

So in 1 hr the distance of separation between them becomes s1+s2 ..that is in effect the speed becomes sum of their speeds.

We say then that the RELATIVE SPEED while traveling in opposite directions is s1+s2

Case 2. They are traveling in same direction with s1>s2 say

In the same way if they travel in the same direction ,their R.S.= s1-s2

Hence in such problems we should use R.S. instead of normal speed in the main
formula I.

RELATIVE DISTANCE :

If A and B are starting from different places or A starts earlier to B and travels some distance before B starts then we should take into account the initial distance of separation and the final distance of separation needed.

RELATIVE DISTANCE TO TRAVEL = (Initial – Final) distance of separation

If they are starting from the same place at the same time their initial distance of separation is zero.

If they are to meet at the end ,then their final distance of separation is zero.

Again ,if there is a difference in time of start , we should account for that and take the time of simultaneous travel only, as shown in some examples below.

Hence the main formula becomes

RELATIVE DISTANCE TO TRAVEL = RELATIVE SPEED * TIME.

We can see the application with a few examples below.More examples can be posted if needed .

EXAMPLES

1.Martina leaves home at 9 A.M., bicycling at a rate of
24mi/h. Two hours later, John leaves, driving at a
rate of 48mi/h. At what time will JOhn catch up with
Martina?
STEPS
1.use relative speed.. this will be difference in
speeds while traveling in same direction.
2.find the difference in time of start if any and its
effect on distance of separation at the start.
3.find the distance of separation at start using given
data and time gap mentioned in 2.
4.note the end criteria..and use standard formula
d=r*t to meet the end criteria using relative speed
for r and difference in distance of separation between
start and end positions.

1.relative speed =48-24=24 mph=r
2.difference in time of start =2 hrs.
3.distance of separation at the start = the head start
martina got in 2 hrs cycling at 24 mph=2*24=48 miles..
4.end criteria is that they meet at the end ..that is
distance of separation at the end is zero.

So difference in distance of separation from start to end
..... =48-0=48=d
so using d=r*t,we get 48=24*t..or..t=48/24=2 hrs.

That is john will catch up with martina in 2 hrs from his
time of start that is 11=00+2....that is 13=00 hrs.
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2. Points A and B are 100 km
apart on a highway. One car starts from A and another

from B at same time. If the cars travel in the same
direction at a constant speed, they meet in 5 hours. If
the cars travel towards each other, they meet in 1
hour. What are the speeds of the two cars?

LET THE SPEEDS OF THE CARS BE X AND Y KMPH
CASE 1.
WHEN GOING IN SAME DIRECTION
RELATIVE SPEED =X-Y
DISTANCE =100 KM
TIME =100/(X-Y)=5 HRS
X-Y=100/5=20.............................I
CASE 2.
WHEN GOING IN OPPOSITE DIRECTION
RELATIVE SPEED =X+Y
TIME =100/(X+Y)=1 HRS
X+Y=100......................II
EQN.I+EQN.II
2X=120
X=120/2=60
Y=100-X=100-60=40
HENCE SPEEDS OF 2 CARS ARE 60 AND 40 KMPH

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