Momentum

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 = \( \Delta \vec P = \vec P_f - \vec P_i \)
          \( = m\vec V_f - m\vec V_i \) \( \left| {\Delta \vec P} \right| = \left| {\vec P_f - \vec P_i } \right| = \sqrt {P_f^2 + P_1^2 - 2P_f P_i \cos \theta } \)
    where \(\theta\) = angle between \( \vec P_f \) and \( \vec P_i \) .

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 = 
                     \( \overrightarrow {\Delta P} = \vec P_f - \vec P_i = 0 - \left( {mv} \right)\hat i \)  
                 \( = - \left( {mv} \right)\hat i \)  \( \left| {\overrightarrow {\Delta P} } \right| = mv \)   
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 \( \theta = 180^0 \) 
                          \( \therefore \left| {\overrightarrow {\Delta P} } \right| = 2mv \)   
     
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
                             \( \left| {\overrightarrow {\Delta P} } \right| = 2mv\,\,\cos \theta \)      

Case(v) : In the above case if \(\theta\) is the angle made by \( \vec V_i \) with wall then \( \overrightarrow {\left| {\Delta P} \right|} = 2mv\,\,\sin \theta \)   

Inertia and Momentum:
Momentum is closely related to the concept of inertia. An object in motion tends to stay in motion (or at rest) due to its inertia. The greater the mass and velocity of an object, the greater its momentum.

Relationship Between Force, Time, and Momentum:
The relationship between force, time, and momentum is expressed by the equation F·t=p    

Real-world Applications:
1. Momentum is crucial in rocketry. According to Newton's Third Law, the expulsion of mass in one direction results in the propulsion of the rocket in the opposite direction.
2. Understanding momentum is essential in traffic safety, as it plays a role in the design of car safety features and the analysis of vehicle collisions.
3. The concept of momentum is applied in analyzing the motion of projectiles, such as calculating the momentum of a launched projectile.