Let's take this example and tweak it a bit to bring it more in line with Relativity. Instead of both cars traveling in the same direction at different speeds, you have two cars traveling at the same speed but is slightly different directions. They both start off from the same point at 100 kph. But in different directions from each other. As each car travels along, it checks its progress against that of the other car, but each car judges this progress as being measured in the direction it itself is driving.
If we label the cars as A and B,this is how things look from A's perspective:
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A makes faster progress in the direction that it s driving than B is making in that same direction.
B sees things like this:
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B sees itself as making more progress than A in the direction that B is traveling.
This is the equivalent of time dilation due to relative motion in Special Relativity. In SR, the two clocks moving relative to each other measure each other as making less "progress through time" than they are, or in other words, aging slower.
Now consider what happens if B, after having traveled for some distance, alters its direction of travel, but not its speed, so that it is heading back towards the path of A.
This is what happens according to A
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B ends up "behind" A. Even if A turns to travel in the same direction as A from this point on, it will always be behind A. With SR, this is the equivalent of a clock traveling away a some high speed and then returning at that speed and showing less time upon returning than the clock that "stayed home". In A's view this happens because B made slower progress during the entire trip.
B sees things a bit differently. Up until it makes it turn, A is "behind" it. But once it turns, A goes from behind to "in front". In the following animation this is shown by having the direction in which B is traveling at an given time as being in the vertical direction. After the turn, A goes from being below B to being above it. After that A still make slower progress than B in the vertical direction, but B never makes up all the distance before it reaches A's path. Again if after this point B turns to follow A it finds itself behind A.
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This illustrate the difference that was alluded to in an earlier post between "time dilation" and "accumulated time difference". Time dilation is the difference in the rate of progress B measures for A at any given point of the trip, which during the two legs has B measuring A's clock as running slower. Accumulated time difference is the what we see at the end of the trip and is the result of the accumulation of the the time dilation during the legs, plus the "jump forward" that A appears to make as seen by B when B makes its turn.
In the animation this looks like a instantaneous jump. in reality it would take some none zero time. If we zoom in on A and B and slow down the turn, this what it would look like to B
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The difference between the Car example and Relativity between clocks, is that with the car example, we are dealing with differences in direction that only involve two spatial dimensions. Relativity deals with space-time, and involve "direction changes" in both the spatial and time dimensions. The cars travel on different paths along in a two dimensional plane, while with SR, clocks traveling at different speeds take different paths through 4 dimensional space-time.
Directly visualizing this 4 dimensional movement is a bit hard to say the least, which is why most people initially struggle with understanding what "causes" one clock to tick off less time than another.