Impulse of a Force

Impulse is a concept related to the effect of an unbalanced force acting on a system. According to Newton's first law of motion, the basic effect of such a force is to change the linear momentum. This relationship is formalized by Newton's second law of motion, which shows that the change in momentum over time is equal to the force. The total change in momentum is given a separate name: Impulse of the force.

The quantity ‘change in momentum’ is separately named as the Impulse of the force.

The relationship between force and the resulting impulse (change in momentum) is explained step-by-step:

1. Start with Newton's Second Law: The force $\vec{F}$ is equal to the rate of change of momentum:
   \[\vec F\] = \[\frac{d\vec{p}}{dt}\]
2. Relate Force and Small Change in Momentum: Rearrange the equation to show that a small change in momentum (\[d\vec{p}\]) is the force acting over a small time (dt):
3. \[d\vec{p}=\vec{F}.dt\]
4. Calculate Total Impulse for Constant Force: If the force (\[\vec{F}\]) is constant and acts for a finite time (t), the total Impulse (\[\vec{J}\]) is calculated by multiplying the force by the time:
5. \[\vec{J}=d\vec{p}=\vec{p}_2-\vec{p}_1=\vec{F}.t\] 
6. Calculate Total Impulse for Varying Force: If the force (\[\vec{F}\]) is not constant but its variation with time is known, the total Impulse (\[\vec{J}\]) is found by integrating the force over the time interval:
7. \[\vec{J}=\Delta\vec{p}=\int d\vec{p}=\int\vec{F}.dt\]

This information relates to what happens when objects collide, where the concept of impulse is most often applied:

• Impact Forces: Colliding objects experience forces along the line of impact that change their momenta.
• Internal Forces: For the system of colliding objects, these forces are internal forces.
• Action-Reaction Pairs: These forces form an action-reaction pair, meaning they are equal and opposite and act on different objects.
• Tangential Force: There is no force along the common tangent (perpendicular to the line of contact).
• Energy Loss in Reality: In reality, part of the mechanical energy (kinetic energy) is lost (converted into non-recoverable forms like sound and heat).
• Total Energy Conservation: The total energy of the system is conserved, even when kinetic energy is lost.
• Relative Velocities (Real World): In reality, the velocity of separation (relative final velocity) is less than the velocity of approach (relative initial velocity) along the line of impact.
• Coefficient of Restitution (e): In reality, the coefficient of restitution is e < 1.
• Elastic Collisions (Ideal): Only during elastic collisions (atomic and molecular level only, never possible in real life) is the kinetic energy conserved, and the velocity of separation is equal to the velocity of approach.