NEWTON’S SECOND LAW OF MOTION

NEWTON’S SECOND LAW OF MOTION
Change of momentum : If ‘u’ and ‘v’ are the initial and final velocity of a body then its, initial  momentum = mu  final momentum = mv
    Now change of momentum = final momentum – initial momentum = mv – mu 
Change in momentum of a body in different cases
    Consider a body of mass m moving with velocity \({\vec V_i}\) and momentum \({\vec P_i}\). Due to a collision (or) due to the action of a force on it suppose its velocity changes to \({\vec V_f}\) and momentum changes to \({\vec P_f}\) in a small time interval \(\Delta t\) .
    Change in momentum of body =\(\Delta \overrightarrow P = \overrightarrow {{P_f}} - \overrightarrow {{P_i}} \)     
                                       \(= m{\vec V_f} - m{\vec V_i}\)         \(\left| {\Delta \vec P} \right| = \left| {{{\vec P}_f} - {{\vec P}_i}} \right| = \sqrt {P_f^2 + P_1^2 - 2{P_f}{P_i}\cos \theta } \)           
    where \(\theta\) = angle between \(\overrightarrow {{P_f}} \) and \(\overrightarrow {{P_i}} \)
Case (i) : Consider a body of mass m moving with velocity \(V\hat i\) . If it hits a rigid surface  (or) a wall and comes to rest.  
Change in momentum of the body = \(\overrightarrow {\Delta P} = {\vec P_f} - {\vec P_i} = 0 - \left( {mv} \right)\hat i\) \(= - \left( {mv} \right)\hat i\)
Note : From law of conservation of linear momentum, theoretically,
Change in momentum of surface / wall = +\(\left( {mv} \right)\hat i\)
Case(ii) : In the above case if the body rebounds with same speed V then \(\theta =180^0\)
 \(\left| {\overrightarrow {\Delta P} } \right| = 2mv\)    
Case (iii) : If a body of mass m moving with velocity \({V_1}\hat i\) hits a rigid wall and rebounds with speed V₂ then \(\theta =180^0\), \(\overrightarrow {\Delta P} = {\vec P_f} - {\vec P_i}\)
\(\left| {\overrightarrow {\Delta P} } \right| = m\left( {{V_2} + {V_1}} \right)\)
Case (iv) : A body of mass m moving with speed V hits a rigid wall at an angle of incidence \(\theta\) and reflects with same speed V  \(\overrightarrow {\Delta P} \) of body is along the normal, away from the wall  \(\left| {\overrightarrow {\Delta P} } \right| = 2mv\,\,\cos \theta \)