Question.
Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following figure is a possible result after collision?
Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following figure is a possible result after collision?
solution:
Answer: Case (ii)
It can be observed that the total momentum before and after collision in each case is constant.
For an elastic collision, the total kinetic energy of a system remains conserved before and after collision.
For mass of each ball bearing m, we can write:
Total kinetic energy of the system before collision:
$=\frac{1}{2} m V^{2}+\frac{1}{2}(2 m) 0$
$=\frac{1}{2} m V^{2}$
Case (i).
Total kinetic energy of the system after collision:
$=\frac{1}{2} m \times 0+\frac{1}{2}(2 m)\left(\frac{V}{2}\right)^{2}$
$=\frac{1}{4} m V^{2}$
Hence, the kinetic energy of the system is not conserved in case (i).
Case (ii)
Total kinetic energy of the system after collision:
$=\frac{1}{2}(3 m)\left(\frac{V}{3}\right)^{2}$
$=\frac{1}{6} m V^{2}$
Hence, the kinetic energy of the system is not conserved in case (iii).
Answer: Case (ii)
It can be observed that the total momentum before and after collision in each case is constant.
For an elastic collision, the total kinetic energy of a system remains conserved before and after collision.
For mass of each ball bearing m, we can write:
Total kinetic energy of the system before collision:
$=\frac{1}{2} m V^{2}+\frac{1}{2}(2 m) 0$
$=\frac{1}{2} m V^{2}$
Case (i).
Total kinetic energy of the system after collision:
$=\frac{1}{2} m \times 0+\frac{1}{2}(2 m)\left(\frac{V}{2}\right)^{2}$
$=\frac{1}{4} m V^{2}$
Hence, the kinetic energy of the system is not conserved in case (i).
Case (ii)
Total kinetic energy of the system after collision:
$=\frac{1}{2}(3 m)\left(\frac{V}{3}\right)^{2}$
$=\frac{1}{6} m V^{2}$
Hence, the kinetic energy of the system is not conserved in case (iii).
