Question

Two particles of mass MA and MB and their velocities are VA and VB respectively collide. After collision they interchange their velocities then $\dfrac{{{M_A}}}{{{M_B}}}$ is?

Answer 1

When two particles collide and there are no other forces acting on the particles, then we can use the concept of conservation of momentum.

Complete step by step solution:
Consider the Initial situation,
The body of mass MA is moving with speed VA, hence its initial momentum can be written as ${P_{AI}} = {\text{ }}{M_A}{V_A}$ .
The body of mass MB is moving with speed VB, hence its initial momentum can be written as ${P_{BI}} = {\text{ }}{M_B}{V_B}$ .
Hence total initial momentum of the system will be
${P_I} = {\text{ }}{P_{AI}} + {\text{ }}{P_{BI}}$ ,
Putting the values from above
${P_I} = {\text{ }}{M_A}{V_A} + {\text{ }}{M_B}{V_B}$ .
Consider the Final situation,
The body of mass MA is moving with speed VB, hence its final momentum can be written as ${P_{AF}} = {\text{ }}{M_A}{V_B}$ .
The body of mass MB is moving with speed VA, hence its final momentum can be written as ${P_{BF}} = {\text{ }}{M_B}{V_A}$ .
Hence total final momentum of the system will be
${P_F} = {\text{ }}{P_{AF}} + {\text{ }}{P_{BF}}$ ,
Putting the values from above
${P_F} = {\text{ }}{M_A}{V_B} + {\text{ }}{M_B}{V_A}$ .
By conservation of momentum,
${P_I} = {P_F}$,
Putting the values from above,
${M_A}{V_A} + {\text{ }}{M_B}{V_B} = {\text{ }}{M_A}{V_B} + {\text{ }}{M_B}{V_A}$,
Taking MA and MB to different sides of the equation
${M_A}{V_A} - {\text{ }}{M_A}{V_B} = {\text{ }}{M_B}{V_A} - {M_B}{V_B}$,
Taking MA and MB common on each side of the equation
${M_A}({V_A} - {\text{ }}{V_B}) = {\text{ }}{M_B}({V_A} - {V_B})$,
Cancelling $({V_A} - {V_B})$ gives
${M_A} = {\text{ }}{M_B}$,
i.e. $\dfrac{{{M_A}}}{{{M_B}}} = 1{\text{ }}$.

Note: Before attempting to solve the problem, the student needs to be able to understand that the conservation momentum is applicable whenever the external force on the system is not present. Another important equation that is used in questions of collision is of the coefficient of restitution, which is applicable for all types of collision, whether external forces are present on the system or not.