11.3.2 Elastic Collisions

In an inelastic collision, some of the mechanical energy is dissipated inside the objects as thermal energy and not all of the kinetic energy is recovered. We’re now interested in “perfect bounce” collisions in which kinetic energy is stored as elastic potential energy in compressed molecular bonds, and then all of the stored energy is transformed back into the post-collision kinetic energy of the objects. A collision in which mechanical energy is conserved is called a perfectly elastic collision. A perfectly elastic collision is an idealization, like a frictionless surface, but collisions between two very hard objects, such as two billiard balls or two steel balls, come close to being perfectly elastic.

A photograph of two balls colliding. — A perfectly elastic collision conserves both momentum and mechanical energy.

FIGURE 11.19 shows a head-on, perfectly elastic collision of a ball of mass $m_{1}$ , having initial velocity $(v_{i x})_{1}$ , with a ball of mass $m_{2}$ that is initially at rest. The balls’ velocities after the collision are $(v_{f x})_{1}$ and $(v_{f x})_{2}$ . These are velocities, not speeds, and have signs. Ball 1, in particular, might bounce backward and have a negative value for $(v_{f x})_{1}$ .

The figure consists of three parts labeled Before, During, and After that illustrate a perfectly elastic collision between two balls labeled 1 and 2.

FIGURE 11.19 Full Alternative Text

The collision must obey two conservation laws: conservation of momentum (obeyed in any collision) and conservation of mechanical energy (because the collision is perfectly elastic). Although the energy is transformed into potential energy during the collision, the mechanical energy before and after the collision is purely kinetic energy. Thus

momentum conservation : m_{1} (v_{f x})_{1} + m_{2} (v_{f x})_{2} = m_{1} (v_{i x})_{1}

(11.27)

energy conservation : \frac{1}{2} m_{1} (v_{f x})_{1}^{2} + \frac{1}{2} m_{2} (v_{f x})_{2}^{2} = \frac{1}{2} m_{1} (v_{i x})_{1}^{2}

(11.28)

Momentum conservation alone is not sufficient to analyze the collision because there are two unknowns: the two final velocities. Energy conservation provides the additional information that we need. Isolating $(v_{f x})_{1}$ in Equation 11.27 gives

(v_{f x})_{1} = (v_{i x})_{1} - \frac{m_{2}}{m_{1}} (v_{f x})_{2}

(11.29)

We substitute this into Equation 11.28:

\frac{1}{2} m_{1} [(v_{i x})_{1} - \frac{m_{2}}{m_{1}} (v_{f x})_{2}]^{2} + \frac{1}{2} m_{2} (v_{f x})_{2}^{2} = \frac{1}{2} m_{1} (v_{i x})_{1}^{2}

With a bit of algebra, this can be rearranged to give

(v_{f x})_{2}_{} [(_{} 1 + \frac{m_{2}}{m_{1}}) (v_{f x})_{2} - 2 (v_{i x})_{1}] = 0

(11.30)

One possible solution to this equation is seen to be $(v_{f x})_{2} = 0$ . However, this solution is of no interest; it is the case where ball 1 misses ball 2. The other solution is

(v_{f x})_{2} = \frac{2 m_{1}}{m_{1} + m_{2}} (v_{i x})_{1}

which, finally, can be substituted back into Equation 11.29 to yield $(v_{f x})_{1}$ . The complete solution is

(v_{f x})_{1} = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} (v_{i x})_{1} (v_{f x})_{2} = \frac{2 m_{1}}{m_{1} + m_{2}} (v_{i x})_{1}

(11.31)

(perfectly elastic collision with ball 2 initially at rest)

Equations 11.31 allow us to compute the final velocity of each ball. These equations are a little difficult to interpret, so let us look at the three special cases shown in FIGURE 11.20.

The figure consists of three cases labeled “a”, b, and c, respectively, that illustrate three cases of special elastic collisions occurring between two balls labeled 1 and 2.

FIGURE 11.20 Full Alternative Text

Case a: $m_{1} = m_{2}$ . This is the case of one billiard ball striking another of equal mass. For this case, Equations 11.31 give

$v_{f 1} = 0 v_{f 2} = v_{i 1}$
Case b: $m_{1} ≫ m_{2}$ . This is the case of a bowling ball running into a Ping-Pong ball. We do not want an exact solution here, but an approximate solution for the limiting case that $m_{1} \to \infty$ . Equations 11.31 in this limit give

$v_{f 1} \approx v_{i 1}_{} v_{f 2} \approx 2 v_{i 1}$
Case c: $m_{1} ≪ m_{2}$ . Now we have the reverse case of a Ping-Pong ball colliding with a bowling ball. Here we are interested in the limit $m_{1} \to 0$ , in which case Equations 11.31 become

$v_{f 1} \approx - v_{i 1} v_{f 2} \approx 0$

These cases agree well with our expectations and give us confidence that Equations 11.31 accurately describe a perfectly elastic collision.

11.3.2 Elastic Collisions

FIGURE 11.19 A perfectly elastic collision.

FIGURE 11.20 Three special elastic collisions.