Two observers A and B study the same particle moving along a straight track. Observer A fixes the origin O at a lamp–post, while observer B chooses another origin O′ which is d metres to the right of O on the same line. The x–coordinates measured by A and the x′–coordinates measured by B are related by x′=x−d. The particle moves from x1 to x2 according to A’s measurements.
a. (i) Express the initial and final coordinates recorded by B in terms of x1, x2 and d.
b. (ii) Derive an algebraic expression for the displacement Δx′ measured by B.
c. (iii) Show explicitly that Δx′ is equal to the displacement Δx measured by A, thereby proving that displacement is independent of the choice of origin.
d. (iv) Verify your result numerically for x1=−5m, x2=+7m and d=+3m.