Question
Asked Nov 22, 2019
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Relative Coordinates In a two-body motion system, verify that the po-
1.
sitions of two particles can be written in terms of the CM and relative positions as
r1 = R+m2r/M and r2 = R - mir/M. Hence confirm that the total KE of the two
particles can be expressed as T = MR2 ur, where u denotes the reduced mass
= m1m2/M and M is the total mass M = m1 + m2.
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Relative Coordinates In a two-body motion system, verify that the po- 1. sitions of two particles can be written in terms of the CM and relative positions as r1 = R+m2r/M and r2 = R - mir/M. Hence confirm that the total KE of the two particles can be expressed as T = MR2 ur, where u denotes the reduced mass = m1m2/M and M is the total mass M = m1 + m2.

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Expert Answer

Step 1
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Consider the diagram for the position vectors of the two particles

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Step 2
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Consider the velocity of the particle 1 be vi, and the velocity of the particle 2 be v2. The velocity of the particles, and the time derivatives of the position vector of the particles are r =1-r2

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Step 3
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Write the expression for the position of the two particles in terms of the centre of mass (CM) of the two-body system, and solve for the relative position of the particle 1 ти + т,, R т, + m, ти + m,(r-r) R М MR %3 (m, + m,)r — т,r %3D Mi — т,r т,r =R+

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Vectors and Scalars

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