Question 8.6: Inelastic collision on an air track Let’s begin with a very ...
Inelastic collision on an air track
Let’s begin with a very straightforward collision on an air track. Suppose we perform a collision experiment using the Velcro®-equipped gliders of Figure 8.10. As in Example 8.4, glider A has a mass of 0.50 kg and glider B has a mass of 0.30 kg; both move with initial speeds of 2.0 m/s. Find the final velocity of the joined gliders, and compare the initial and final kinetic energies.
SET UP Figure 8.12 shows our sketches for this problem. As in Example 8.4, we point the x axis in the direction of motion. All the velocity and momentum vectors lie along the x axis.
SOLVE We need to write expressions for the total x component of momentum before and after the collision and equate them. From conservation of the x component of momentum, we have
(0.50 kg)(2.0 m/s)+(0.30 kg)(-2.0 m/s)
=(0.50 kg +0.30 kg)(\upsilon _{f,x}),
\upsilon_{f,x}=0.50 m/s.
Because \upsilon_{f,x} is positive, the gliders move together to the right (the +x direction) after the collision. Before the collision, the kinetic energy of glider A is
K_{A,i}=\frac{1}{2}m_A(\upsilon _{A,i,x})^2=\frac{1}{2}(0.50 kg)(2.0 m/s)^2 =1.0 J,
and that of glider B is
K_{B,i}=\frac{1}{2}m_B(\upsilon _{B,i,x})^2=\frac{1}{2}(0.30 kg)(-2.0 m/s)^2 =0.60 J
Note that the initial kinetic energy of glider B is positive, even though the x components of its initial velocity, \upsilon _{B,i,x}, and momentum, m_B(\upsilon _{B,i,x}) are both negative. (Remember that kinetic energy is a scalar, not a vector component, and that it can never be negative.) The total kinetic energy before the collision is 1.6 J. The kinetic energy after the collision is
\frac{1}{2}(m_A + m_B)(\upsilon _{f,x})^2=\frac{1}{2}(0.50 kg + 0.30 kg)(0.50 m/s)^2
= 0.10 J.
REFLECT The final kinetic energy is only \frac{1}{16} of the original, and \frac{15}{16} is “lost” in the collision. Of course, it isn’t really lost; it is converted from mechanical energy to various other forms of energy. For instance, if there were a wad of chewing gum between the gliders, it would squash irreversibly on impact and become warmer. If the gliders coupled together like two freight cars, the energy would go into elastic waves that would eventually dissipate. If there were a spring between the gliders that compressed as they locked together, then the energy would be stored as potential energy in the spring. In all of these cases, the total energy of the system is conserved, even though kinetic energy is not. However, in an isolated system, momentum is always conserved, whether the collision is elastic or not.
Practice Problem: If glider A has twice the mass of glider B, determine the ratio of final to initial kinetic energy. Answer: \frac{1}{9} .
