Cars on a Straight Road
Cars A, B, C, and D are on a straight road (not necessarily at the same point). They each drive at a constant speed (which is possibly different from each other).
Car A meets
- car B at 10 am
- car C at 1 pm
- car D at 4 pm.
Car D meets
- car B at 12 pm
- car C at 2 pm
- car A at 4 pm (as stated earlier).
What time do car B and car C meet?
Hint:
There is a nice way to represent this problem.
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1 solution
[See if the hint can help you approach this problem, before reading the full solution below.]
Hint: Draw the location-time graph.
Each car will be represented by a straight line because it travels at a constant speed.
What can we say about the lines representing cars B and C?
On the location-time graph, draw lines A and D to represent the path taken by each car.
Mark out points "AB, AC, AD, BD, BC" at the given times on the given lines, to represent when any 2 of these cars meet.
We know car B is at points AB and BD, and travels in a straight line, so the line representing car B is formed by connecting AB and BD.
Similarly, the line representing car C is formed by connecting AC and CD.
Now, observe that triangles "AD-CD-AC" and "AD-BD-AB" are similar in the ratio 1:2, and so line B is parallel to line C.
This tells us that B and C will never meet (and also that they have the same speed+direction).
Here is an example of what the location-time graph might look like with cars A and D are traveling in opposite directions.
Note: Is it possible that A, B, C, D are all driving in the same direction?
If yes, what additional information can we conclude?
Can you come up with such a graph to show that this scenario is possible?
Note: Can you draw the graphs representing these scenarios:
1) A+D in the same direction with A faster,
2) A+D in same directions with D faster,
3) A stationary,
4) D stationary.
This shows us that we cannot draw any conclusions about the relative speed / direction of cars A and D. They could have been any 2 (non-vertical) lines that we drew on the plane to intersect once at 4pm (at some location). In fact, one of them could have been a horizontal line (meaning that the car is traveling at a constant speed of 0).