Answer to Question #142698 in Mechanics | Relativity for Mac Evans

Question

A 3-kg object moving to the right on a frictionless, horizontal surface with a speed of 2 m/s collides head on and sticks to a 2-kg object that is initially moving to the left with a speed of 4 m/s. After the collision, which statenoent is true?
(a) The kinetic energy of the system is 20 J. (b) The momentum of the system is 14 kgm/s. (c)
The kinetic energy of the system is greater than 5 J but less than 20 J. (d) The momentum of the system is 22?

Answer 1

The kinetic energy of the system is (assuming that all kinetic energy of both objects after collision converts into kinetic energy of the system):

$E=mv^2/2=m_1v_1^2/2+m_2v_2^2/2,$ (1)

where m - the mass of the system, that equal $m_1+m_2$ ;

$v$ - the speed of the system;

$m_1$ - the mass of the first object;

$v_1$ - the speed of the first object;

$m_2$ - the mass of the second object;

$v_2$ - the speed of the second object.

$E=3\cdot 2^2/2+2\cdot 4^2/2=22\space J.$

The momentum of the system is 

$P=mv=(m_1+m_2)v.$

From (1)

$v=\sqrt {\frac {m_1v_1^2+m_2v_2^2}{m_1+m_2}}=\sqrt {\frac {3\cdot 2^2+2\cdot 4^2}{3+2}}=2.97\space m/s.$

$P=(2+3)\cdot 2.97=14.83\space kg\cdot m/s.$

Answer: (b) The momentum of the system is 14 kgm/s (approximately!!!).