Having trouble visualizing something about acceleration

[Byrne]

Okay, so say one car accelerates for 20 seconds at a speed of 1.35 m/s² while a second car accelerates for 10 seconds at a speed of 2.70 m/s². Notice that the second car accelerates twice as quickly, but in half the time. Now my question lies in describing the displacement of these two cars. They will both reach the same final velocity, assuming all accelerations stops after the aforementioned time intervals. If you were to graph these intervals, it is pretty easy to see that the second car travels half the distance as the first car in the 10 second interval. I know this is true because of the graph, but for some reason I am having troubles visualizing this. Could anyone may paint a word picture for me to help better my visualizing of this. It may seem pretty simple, but I'd rather be able to fully understand rather than just memorize.

 

[andrevdh]

Have a look at the simulations at http://wps.aw.com/aw_young_physics_11" . Click on the icons in the text to open the simulations. You need Macromedia Flash player 6.0.65 plugin installed in order to run the simulations.

 

[Doc Al]

One thought that might help is to realize that both cars have the same average speed ($v_{ave} = (v_i + v_f)/2$), but since the first car moves for twice the time it goes twice as far.