Perfectly Inelastic Collision
Category : JEE Main & Advanced
In such types of collisions, the bodies move independently before collision but after collision as a one single body.
(1) When the colliding bodies are moving in the same direction
By the law of conservation of momentum
\[{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}=({{m}_{1}}+{{m}_{2}}){{v}_{\text{comb}}}\]
\[\Rightarrow \] \[{{v}_{\text{comb}}}=\frac{{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}}{{{m}_{1}}+{{m}_{2}}}\]
Loss in kinetic energy
\[\Delta K=\left( \frac{1}{2}{{m}_{1}}u_{1}^{2}+\frac{1}{2}{{m}_{2}}u_{2}^{2} \right)-\frac{1}{2}({{m}_{1}}+{{m}_{2}})v_{comb}^{2}\]
\[\Delta K=\frac{1}{2}\left( \frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}} \right)\,{{({{u}_{1}}-{{u}_{2}})}^{2}}\]
[By substituting the value of \[{{\upsilon }_{comb}}\]]
(2) When the colliding bodies are moving in the opposite direction
By the law of conservation of momentum
\[{{m}_{1}}{{u}_{1}}+{{m}_{2}}(-{{u}_{2}})=({{m}_{1}}+{{m}_{2}}){{v}_{\text{comb}}}\]
(Taking left to right as positive)
\[\therefore \] \[{{v}_{\text{comb}}}=\frac{{{m}_{1}}{{u}_{1}}-{{m}_{2}}{{u}_{2}}}{{{m}_{1}}+{{m}_{2}}}\]
when \[{{m}_{1}}{{u}_{1}}>{{m}_{2}}{{u}_{2}}\] then \[{{v}_{\text{comb}}}>0\] (positive)
i.e. the combined body will move along the direction of motion of mass \[{{m}_{1}}\].
when \[{{m}_{1}}{{u}_{1}}<{{m}_{2}}{{u}_{2}}\] then \[{{v}_{\text{comb}}}<0\] (negative)
i.e. the combined body will move in a direction opposite to the motion of mass \[{{m}_{1}}\].
(3) Loss in kinetic energy
\[\Delta K=\] Initial kinetic energy - Final kinetic energy
\[=\left( \frac{1}{2}{{m}_{1}}u_{1}^{2}+\frac{1}{2}{{m}_{2}}u_{2}^{2} \right)-\left( \frac{1}{2}({{m}_{1}}+{{m}_{2}})\,v_{\text{comb}}^{2} \right)\]
\[=\frac{1}{2}\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{({{u}_{1}}-{{u}_{2}})}^{2}}\]
You need to login to perform this action.
You will be redirected in
3 sec
