Choose your own velocity

When solving physics problems we always have the benefit of choosing our frame of reference.  Not only can we chose where the origin of our coordinate is, but we can also chose which part of our system is stationary.  Consider the following simple kinematic problem:

A car is travelling along, and we want to know how long it takes to travel some distance.  We can plug numbers into a kinematic equation and get the answer.  If the car is accelerating then the problem is mathematically a bit more fiddly (a quadratic equation, or do it in two steps with two equations) but it's not conceptually harder.

But now consider a similar problem with two moving bodies:

Now there's two cars at different speeds, and the question is: how long will it take the red car to overtake the blue car?  This looks rather harder because the distance the red car has to travel is not immediately clear.  But that's only because of our choice of reference frame, which up till now has been implicit.

We instinctively chose a frame in which the ground, represented in the diagram by a tree, is stationary.  We might call this the "Earth frame".  A better diagram would make that explicit:

The problem is made much easier if we choose a different frame, one in which the blue car is stationary.  Simple addition of velocities gives us:

The velocities in the new "blue car frame" are marked with a prime.  The distance to be covered by the red car is now clear, because the blue car is not moving.

To complete the example: suppose the red car is 20 m behind the blue car; the red car is 9 m long (longer than the blue car) and the driver wants to be 5 m past the blue car.  That makes for a total distance of 34 m.  To ensure the maneouvre is completed in reasonable time, the driver also accelerates at 2 m/s/s.

(I have neglected the extra distance required for the red car to pull out from behind the blue, then return to the lane in front of the blue car.)

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