Section 7.8 The CM Frame
Consider a block A moving towards another block B at rest on a table as shown in Figure 7.8.1. The left side of the figure shows the drawing in standard frame, called LAB frame. The CM of the two blocks also moves towards B here, but at a lower speed than speed of A.

The first row in Figure 7.8.1 is the situation before the collision and the bottom part is the situation after the collision. The left two figure illustrate that momentum is conserved in the collision; the total momentum given by total mass times the CM velocity does not change in collision.
The right two figures in Figure 7.8.1 show the same situation in the CM frame. In the CM frame, the origin is at the CM. In this frame, both A and B have non-zero velocity. Before the collision, they are moving towards origin with momentum of equal magnitude and opposite direction.
That is,
After collision, they are moving away from CM, again with momentum of equal magnitude and opposite direction.
Because of the symmetry in the before and after collision, as observed in the CM frame, this frame is often used to study collision problems.
Subsection 7.8.1 Relation between CM and LAB frames
Relating LAB and CM frames is no different than relating two frames. Suppose we want to find the relation for position, velocity and acceleration between LAB and CM frames.
Let \(\vec R\) be the position of the origin of the CM frame with respect to the origin of the LAB frame. Let \(\vec r\) and \(\vec r_\text{cm}\) be position of a particle with respect to the origins of the LAB and CM frames respectively. Then, from traingle of vectors we will have

Here, \(\vec r\) and \(\vec R\) are in the LAB frame and \(\vec r_\text{cm}\) is in the CM frame. Now, by taking a time derivative we immediately get the relation among velocities.
Another derivative gives us relation among the accelerations.
Subsection 7.8.2 Velocities in CM and LAB Frame of Two Bodies
Let \(m_A\) and \(m_B\) be the masses of two bodies A and B with velocities \(\vec v_{A,\text{cm}}\) and \(\vec v_{B,\text{cm}}\) in the CM frame and \(\vec v_{A,\text{LAB}}\) and \(\vec v_{B,\text{LAB}}\) in the LAB-frame. In the LAB frame, we can easily see that the velocity of CM itself with respect to the LAB frame, to be denoted as \(\vec V\text{,}\) will be
Now, we can use Eq. (7.8.2), to write the relation between LAB and CM velocities of the two bodies.
Subsection 7.8.3 Collision in CM Frame
Figure 7.8.1 shows a one-dimensional collision in LAB and CM frames. There we find that in the CM frame, both particles appraoch the CM with equal magnitude momentum and after collision move away with some other magnitude momentum, again both with equal magnitude.
In more than two or more dimensions, the directions of the menenta of the two bodiesare still along the same line, but the line of the motions of the two bodies may be different than the line they approached the CM before the collision. That is, there is a rotation of the direction of motion in the collision process as shown in Figure 7.8.3.

In Figure 7.8.3, we will have the following relation for the conservation of momentum in the collision process, noting that net momentum in CM frame is zero.
The angle of rotation of direction can be obtained from looking at the ratio of \(x\) and \(y\) components of \(\vec v'_{A,\text{cm}}\) or \(\vec v'_{B,\text{cm}}\text{.}\)
Checkpoint 7.8.4. Scattering of Alpha Particles from Gold Nucleus.
Alpha particles of mass \(4 \text{AMU}\) are incident on gold nucleus of mass \(197\text{ AMU}\text{.}\) Before the collision, alpha particle is moving with a speed of \(2\times10^5\ \text{m/s}\) in the LAB-frame. After the collision, the alpha particle comes out with a speed of \(1.5\times10^5\ \text{m/s}\) at an angle of \(10^{\circ}\) from the original direction in the LAB frame. Find the angle between the incoming and outgoing directions in the CM-frame. AMU is atomic mass unit with value \((1\ \text{AMU} = 1.66053\times10^{-27} \text{ kg})\text{.}\)
First use conservation of momentum in LAB frame.
\(10^\circ\text{.}\)
Let us orient axes of the LAB and CM frames so that their axes are parallel, and initially the motion is along the \(x\)-axis. The velocity of the origin of the CM-frame with respect to the LAB-frame will be towards positive \(x\)-axis. Only \(V_x\) will be non-zero.
Let \(v_x\) and \(v_y\) denote the velocity (in LAB frame) of gold nucleus after collision. Conservation of momentum in LAB frame gives
Therefore,
Now, we can compute velocities of alpha particle and gold nucleus in CM frame by subtracting the velocity of CM from these numbers.
Therefore, angle between incoming and outgoing directions is
or,