This is simplest motion possible for "two body" system. The bodies under consideration are initially at rest. In this case, center of mass of two bodies is a specific point in the given reference. Also, it is to be noted that center of mass lies always between two bodies and not beyond them.

Since no external force is applied, the subsequent motion due to internal gravitational force does not change the position of center of mass in accordance with second law of motion. The bodies simply move towards each other such that center of mass remains at rest.

The two bodies move along a straight line joining their centers. The line of motion also also passes through center of mass. This has one important implication. The plane containing one body and “center of mass” and the plane containing other body and “center of mass” are same. It means that motions of two bodies are “coplanar” with line joining the centers of bodies and center of mass. The non-planar motions as shown in the figure below are not possible as motion is not along the line joining the centers of two bodies.

Figure 2: The motion of two body system can not be non-planar.

Two body system

Let suscripts "1" and "2" denote two bodies. Also, let “ r 1 r 1 ”, “ v 1 v 1 ”, “ a 1 a 1 ” and “ r 2 r 2 ”, “ v 2 v 2 ”, “ a 2 a 2 ” be the magnitudes of linear distance from the center of mass, speeds and magnitudes of accelerations respectively of two bodies under consideration. Also let “center of mass” of the system is the origin of reference frame. Then, by definition of center of mass :

Figure 3: Center of mass is the origin of reference frame.

Two body system

r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2 r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2

But “center of mass” lies at the origin of the reference frame,

⇒ r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2 = 0 ⇒ r c m = - m 1 r 1 + m 2 r 2 m 1 + m 2 = 0

⇒ m 1 r 1 = m 2 r 2 ⇒ m 1 r 1 = m 2 r 2

Taking first differentiation of position with respect to time, we have :

⇒ v c m = - m 1 v 1 + m 2 v 2 m 1 + m 2 = 0 ⇒ v c m = - m 1 v 1 + m 2 v 2 m 1 + m 2 = 0

⇒ m 1 v 1 = m 2 v 2 ⇒ m 1 v 1 = m 2 v 2

Taking first differentiation of velocity with respect to time, we have :

⇒ a c m = - m 1 a 1 + m 2 a 2 m 1 + m 2 = 0 ⇒ a c m = - m 1 a 1 + m 2 a 2 m 1 + m 2 = 0

⇒ m 1 a 1 = m 2 a 2 ⇒ m 1 a 1 = m 2 a 2

Considering only magnitude and combining with Newton’s law of gravitation,

F 12 = F 21 = G m 1 m 2 r 1 + r 2 2 F 12 = F 21 = G m 1 m 2 r 1 + r 2 2

Since distance of bodies from center of mass changes with time, the gravitational force on two bodies is equal in magnitude at a given instant, but varies with time.

We can treat “two body” system equivalent to “one body” system by stating law of motion in appropriate terms. For example, it would be interesting to know how force can be related to the relative acceleration with which two bodies are approaching towards each other. Again, we would avoid vector notation and only consider the magnitudes of accelerations involved. The relative acceleration is sum of the magnitudes of individual accelerations of the bodies approaching towards each other :

a r = a 1 + a 2 a r = a 1 + a 2

According to Newton's third law, gravitational force on two bodies are pair of action and reaction and hence are equal in magnitude. The magnitude of force on each of the bodies is related to acceleration as :

F = m 1 a 1 = m 2 a 2 F = m 1 a 1 = m 2 a 2

⇒ m 1 a 1 = m 2 a 2 ⇒ m 1 a 1 = m 2 a 2

We can note here that this relation, as a matter of fact, is same as obtained using concept of center of mass. Now, we can write magnitude of relative acceleration of two bodies, “ a r a r ”, in terms of individual accelerations is :

⇒ a r = a 1 + m 1 a 1 m 2 ⇒ a r = a 1 + m 1 a 1 m 2

⇒ a r = a 1 m 1 + m 2 m 2 ⇒ a r = a 1 m 1 + m 2 m 2

⇒ a 1 = m 2 a r m 1 + m 2 ⇒ a 1 = m 2 a r m 1 + m 2

Substituting in the Newton’s law of motion,

⇒ F = m 1 a 1 = m 1 m 2 a r m 1 + m 2 ⇒ F = m 1 a 1 = m 1 m 2 a r m 1 + m 2

⇒ F = μ a r ⇒ F = μ a r

Where,

⇒ μ = m 1 m 2 m 1 + m 2 ⇒ μ = m 1 m 2 m 1 + m 2

Velocity of approach

The two bodies are approaching towards each other. Hence, magnitude of velocity of approach is given by :

Figure 4: Two bodies are approaching each other along a straight line.

Velocity of approach

v r = v 1 + v 2 v r = v 1 + v 2

This is the expression of the magnitude of velocity of approach. We can write corresponding vector equation for velocity of approach in relation to reference direction. In this module, however, we will avoid vector notation or vector interpretation to keep the discussion simplified.

The kinetic energy of the “two body” system is given as the sum of kinetic energy of individual bodies,

K = 1 2 m 1 v 1 2 + 1 2 m 2 v 2 2 K = 1 2 m 1 v 1 2 + 1 2 m 2 v 2 2

We can write this expression of kinetic energy in terms of relative velocity i.e. velocity of approach. For this, we need to express individual speeds in terms of relative speed as :

v r = v 1 + v 2 v r = v 1 + v 2

But,

m 1 v 1 = m 2 v 2 m 1 v 1 = m 2 v 2

⇒ v 2 = m 1 v 1 m 2 ⇒ v 2 = m 1 v 1 m 2

Substituting in the expression of relative velocity,

⇒ v r = v 1 + m 1 v 1 m 2 ⇒ v r = v 1 + m 1 v 1 m 2

⇒ v 1 = m 2 v r m 1 + m 2 ⇒ v 1 = m 2 v r m 1 + m 2

Similarly,

⇒ v 2 = m 1 v r m 1 + m 2 ⇒ v 2 = m 1 v r m 1 + m 2

Now, putting these expressions of individual speed in the equation of kinetic energy :

⇒ K = m 1 m 2 2 v r 2 m 1 + m 2 2 + m 2 m 1 2 v r 2 m 1 + m 2 2 ⇒ K = m 1 m 2 2 v r 2 m 1 + m 2 2 + m 2 m 1 2 v r 2 m 1 + m 2 2

⇒ K = m 1 m 2 v r 2 m 1 + m 2 2 m 1 + m 2 ⇒ K = m 1 m 2 v r 2 m 1 + m 2 2 m 1 + m 2

⇒ K = m 1 m 2 v r 2 m 1 + m 2 ⇒ K = m 1 m 2 v r 2 m 1 + m 2

⇒ K = 1 2 μ v r 2 ⇒ K = 1 2 μ v r 2

This result also indicates that we can treat “two body” system as “one body” system from the point of view of kinetic energy, as if the body has reduced mass of “μ” and speed equal to the magnitude of relative velocity, “ v r v r ”.

From the discussion above, we conclude the followings about the motion of “two body” system along a straight line :

1: Each body follows a straight line trajectory.

2: The line joining centers of two bodies pass through center of mass.

3: The planes of two motions are in the same plane. In other words, two motions are coplanar.

4: Magnitude of gravitational force is same for two bodies, but they vary as the distance between them changes.

5: We can treat “two body” system equivalent to “one body” system by using concepts of (i) reduced mass “μ” (ii) relative velocity, “ v r v r ” and (iii) relative acceleration “ a r a r ”.