Distance Rate Time Dilemma

Question

1.) Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?

2.) miami and orlando are 210 miles apart. a truck traveled from miami toward orlando at the rate of 48 mph. another truck traveled from orlando toward miami at the rate of 42 mph. both trucks started traveling at the same time. how many miles did each travel before they met?

The above questions, I note assume that both cyclists and the trucks travel for the same amount of time before they meet.

No where in the question is this mentioned and I fail to understand why we are assuming that if they leave at the same time, they must also travel for the same amount of time before they meet.

Could someone clarify this please?

Thanks

fivedaysleft · Answer

in both questions the time travelled is t = total distance/(sum of speeds)

your question is
"No where in the question is this mentioned and I fail to understand why we are assuming that if they leave at the same time, they must also travel for the same amount of time before they meet."

think of it this way.suppose you and your friend live down the same road. you place a phone call and say we need to meet up. urgent. both of you leave your respective houses at the same time...and start walking towards each other's house. after 10 minutes you both meet. so the time elapsed for either of you is 10 minutes. whether he limps, or you come by a car....if the time in your watch is 10 minutes, thats what his watch also says.
this time will be t = total distance between your houses/ (sum of your speeds)
if you come by a car, your speed will be much higher and so you will cover more distance...and if you are limping your speed will be slower and your friend will cover more distance...BUT BOTH YOUR WATCHES WILL SHOW THE SAME TIME ELAPSED.

hope i make myslef clear?

sudhir18n · Answer

when 2 objects are travelling in opposite direction towards each other .. their relative speed= sum of their speeds.

Q1. total distance 45 miles
 relative speed = 14+16 = 30 
hence time 45/30 = 1.5 hrs

Q2: same approach total distance = 210 miles
 relative speed = 48+42 = 90 
thus time = 210/90 = 7/3 = 2 hrs. 20 mints 
hence first truck travels @ 48mph .. hence in 2hrs 20 mints it would travel 112 miles ( 96 + 48/60*20)
second truck @ 42 mph = 84+ 42/60*20 = 84+ 14 = 98 miles

mist3rh · Answer

Thanks guys, but I remain confused.

1.) I understand that relative motion in opp direction, relative velocity being sum of individual speeds.
But why are we using time = distance/sum of their speeds ? Time = Distance/Speed for a given object in motion. Here we are talking about 2 different objects. How can we simply add their speeds ?

2.)fivedaysleft, Yes we could start at the same time say 9 am and walk towards each other. We meet at a given point X. Obviously, we will have covered different distances at different speeds, but the time taken to cover the distances will also be different.
Total distance = 100
Distance 1 : 60 = 6x 10
Distance 2 : 40 = 8 x 5

As you can see, that in above example, different distances and different speeds have different times. So the assumption that they take same time is incorrect.

fivedaysleft · Answer

for 2.)fivedaysleft, Yes we could start at the same time say 9 am and walk towards each other. We meet at a given point X. Obviously, we will have covered different distances at different speeds, but the time taken to cover the distances will also be different.
Total distance = 100
Distance 1 : 60 = 6x 10
Distance 2 : 40 = 8 x 5

the distance covered will be different as you said, but it will be in the ratio of their speeds...
speed(object1) is propotional to distance(object 1)
speed(object2) is propotional to distance(object 2)

or speed1/ speed2 equals distance1/distance 2 = 60/40
and the time remains constant...

now as per 1.) you have to know that relative velocity is v(object1)-v(object2)
consider : a----------------------------------------------------b
 d

a moves right to left and b moves left to right ... they cover the distance d
now chose one direction as positive....say left to right
then Va=+v(objecta)
and Vb = -v(objectb) since it is in the opposite direction
so, relative velocity will be Va-Vb = +v(objecta) -   = v(objecta)+v(objectb) ie sum of speeds

if you take right to left as positive ie "B" reference relative velocity will be +v(objectb) -   which gives v(objectb)+v(objecta) ie sum of speeds

hope it clears your doubt

mist3rh · Answer

fivedaysleft, I agree the distance will be proportional to their speeds SHOULD the time be constant. However, that is exactly what my point is. We do not know whether the time is constant. As I put an example up before, for 2 different distances 60 and 40, at different speeds 6 & 8, the time has to be 10 & 5.Clearly the times are NOT same and hence NOT constant. So unless specifically mentioned in the question, we do not know whether the time is same. But answers the both of you gave, you simply assumed the time is same/constant.

I understand that relative velocity will be sum of speeds -question being, why are we calculating time using sum of their speeds. THese are 2 different objects. and the formula is true when specific to an object i.e. for each truck or cycle, D= S x T. We can't just add D= (s1+s2) x T. Thats incorrect.

Distance Rate Time Dilemma

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