Question

A body of mass 2 kg moving with velocity 3 m/s collides with a body of mass 1 kg moving with a velocity of 4 m/s in the opposite direction. If the collision is head on a completely inelastic, then:
A. both particles move together with velocity ($\dfrac{2}{3}$) m/s
B. the momentum of system is throughout
C. the momentum of system is 10 kg m/s
D. the loss of KE of system is ($\dfrac{{49}}{3}$) J

Answer 1

Hint First of all, combine the mass of both the bodies then apply the law of conservation of momentum to calculate the velocity of both the bodies. Calculate their momentum and find their difference. Then calculate the loss in kinetic energy of the system due to collision.

Complete step-by-step solution:
We need to find the mass of two bodies as they are moving in opposite direction:
Let, ${m_1}$= mass of 1st body
       ${m_2}$= mass of 2nd body
       ${u_1}$= velocity of 1st body
       ${u_2}$= velocity of 2nd body
       $P$ = Momentum of system
According to the question it is given that,
${m_1}$= 2 kg ${m_2}$= 1 kg ${u_1}$= 3 m/s ${u_2}$=4 m/s
Let the common velocity of combined body be V and common mass be M
$M = 2 + 1 = 3kg$
Now, applying conservation of momentum which can be stated as: - “For a collision occurring between object 1 and object 2 in an isolated system, the total momentum of the two objects before the collision is equal to the total momentum of the two objects after the collision. That is, the momentum lost by object 1 is equal to the momentum gained by object 2”.
$2 \times 3 - 1 \times 4 = 3 \times V$
$ \Rightarrow V = \dfrac{2}{3}m/s$
Momentum of system, $P = 2 \times 3 - 1 \times 4 = 2kgm/s$
Loss in kinetic energy of system due to collision
$ \Rightarrow \vartriangle K.E = K.{E_i} - K.{E_f}$
Kinetic Energy $K.E = \dfrac{1}{2}M{V^2}$
$\vartriangle K.E = \dfrac{1}{2}\left( 2 \right) \times {3^2} + \dfrac{1}{2}\left( 1 \right) \times {4^2} - \dfrac{1}{2}\left( 3 \right) \times {\left( {\dfrac{2}{3}} \right)^2}$
$ \Rightarrow \vartriangle K.E = \dfrac{{49}}{3}J$

Hence, there is a loss of $\dfrac{{49}}{3}J$ Kinetic Energy
Therefore, option (D) is correct.

Note:- The mass and velocity of both bodies should be combined and then apply the law of conservation of momentum. This is to calculate the combined velocity of two bodies.