Sample Problem 9-6

Figure 9-9a shows a space hauler and cargo module, of total mass traveling along an axis in deep space. They have an initial velocity of magnitude 2100 km/h relative to the Sun. With a small explosion, the hauler ejects the cargo module, of mass 0.20M (Fig. 9-9b). The hauler then travels 500 km/h faster than the module along the x axis; that is, the relative speed between the hauler and the module is 500 km/h. What then is the velocity of the hauler relative to the Sun?

SOLUTION: The Key Idea here is that, because the hauler-module system is closed and isolated, its total linear momentum is conserved; that is,

, (9-31)

where the subscripts and refer to values before and after the ejection, respectively. Because the motion is along a single axis, we can write momenta and velocities in terms of their components. Before the ejection, we have

(9-32)

Let be the velocity of the ejected module relative to the Sun. The total linear momentum of the system after the ejection is then

(9-33)

where the first term on the right is the linear momentum of the module and the second term is that of the hauler.

We do not know the velocity of the module relative to the Sun, but we can relate it to the known velocities with

In symbols, this gives us

Or

Substituting this expression for into Eq. 9-33, and then substituting Eqs. 9-32 and 9-33 into Eq. 9-31, we find

, which gives us

or

km/h. (Answer)