Linear Momentum of System — Revision Notes

⚡ 30-Second Revision

Linear Momentum: — $vec{p} = mvec{v}$ (vector, unit kg·m/s)
Total System Momentum: — $vec{P}_{sys} = sum vec{p}_i = Mvec{v}_{CM}$
Newton's 2nd Law for System: — $vec{F}_{ext} = \frac{dvec{P}_{sys}}{dt}$
Conservation of Momentum: — If $vec{F}_{ext} = 0$, then $vec{P}_{sys} = \text{constant}$
Impulse: — $vec{J} = int vec{F} dt = Deltavec{p}$
Collisions:

- Elastic: Momentum conserved, Kinetic Energy conserved. - Inelastic: Momentum conserved, Kinetic Energy NOT conserved. - Perfectly Inelastic: Objects stick together, momentum conserved, max KE loss.

2-Minute Revision

Linear momentum, $vec{p} = mvec{v}$, is a vector quantity representing 'mass in motion'. For a system of particles, the total linear momentum $vec{P}_{sys}$ is the vector sum of individual momenta, or equivalently, the product of total mass and the velocity of the center of mass ($Mvec{v}_{CM}$).

The crucial principle is the Law of Conservation of Linear Momentum: if the net external force ($vec{F}_{ext}$) acting on a system is zero, then its total linear momentum remains constant. Internal forces, acting between particles within the system, do not change the total momentum; they only redistribute it.

This law is fundamental for analyzing interactions like collisions, explosions, and recoil, where external forces are often negligible compared to large internal forces. Remember to treat momentum as a vector, carefully assigning directions (signs in 1D, components in 2D).

The impulse-momentum theorem ($vec{J} = Deltavec{p}$) is used when force acts over a time interval, especially for time-varying forces requiring integration.

5-Minute Revision

Linear momentum ($vec{p} = mvec{v}$) is a vector quantity, crucial for understanding dynamics. For a system, total momentum $vec{P}_{sys} = sum m_ivec{v}_i = Mvec{v}_{CM}$. The rate of change of this total momentum is equal to the net external force: $vec{F}_{ext} = \frac{dvec{P}_{sys}}{dt}$.

The cornerstone is the Law of Conservation of Linear Momentum: if $vec{F}_{ext} = 0$, then $vec{P}_{sys}$ is constant. This means initial total momentum equals final total momentum. Internal forces (e.

g., forces in a collision) cancel out in pairs and do not affect the *total* system momentum, only its distribution among particles.

Applications:

1
Collisions: — Always conserve momentum. For 1D, use signs for direction: $m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2'$. For 2D, conserve momentum along x and y axes independently: $P_{x,initial} = P_{x,final}$ and $P_{y,initial} = P_{y,final}$. Remember, kinetic energy is conserved *only* in elastic collisions.
2
Recoil: — In a gun-bullet system, initial momentum is zero. After firing, $0 = m_{bullet}vec{v}_{bullet} + m_{gun}vec{v}_{gun}$. The gun recoils opposite to the bullet.
3
Explosions: — An object at rest explodes into fragments. The vector sum of momenta of all fragments must be zero. If initially moving, the initial momentum equals the vector sum of final momenta.

Impulse-Momentum Theorem: Impulse $vec{J} = int vec{F} dt = vec{F}_{avg}Delta t$. It also equals the change in momentum: $vec{J} = Deltavec{p} = vec{p}_{final} - vec{p}_{initial}$. This is useful for impact forces.

Key Strategy: Always define your system, identify external forces, and apply vector addition carefully. Don't confuse momentum with kinetic energy.

Prelims Revision Notes

1
Linear Momentum ($vec{p}$): — Product of mass and velocity. $vec{p} = mvec{v}$. It's a vector quantity, direction same as $vec{v}$. SI unit: kg·m/s or N·s.
2
Total Linear Momentum of a System ($vec{P}_{sys}$): — Vector sum of individual momenta. $vec{P}_{sys} = sum_{i} m_ivec{v}_i$. Also, $vec{P}_{sys} = Mvec{v}_{CM}$, where $M$ is total mass and $vec{v}_{CM}$ is velocity of center of mass.
3
Newton's Second Law for a System: — The net external force on a system equals the rate of change of its total linear momentum. $vec{F}_{ext} = \frac{dvec{P}_{sys}}{dt}$.
4
Internal vs. External Forces:

* Internal Forces: Act between particles within the system. They always cancel out in pairs (Newton's 3rd Law) and do NOT change the *total* linear momentum of the system. * External Forces: Act on the system from outside. Only these can change the total linear momentum of the system.

1
Law of Conservation of Linear Momentum: — If the net external force acting on a system is zero ($vec{F}_{ext} = 0$), then the total linear momentum of the system remains constant ($vec{P}_{sys} = \text{constant}$). This is a vector conservation law, meaning components along independent axes are conserved if the corresponding external force components are zero.
2
Applications:

* Collisions (1D & 2D): Total momentum is always conserved in an isolated system. $m_1vec{v}_1 + m_2vec{v}_2 = m_1vec{v}_1' + m_2vec{v}_2'$. For 2D, resolve into x and y components. Kinetic energy is conserved only in elastic collisions.

* Perfectly Inelastic Collisions: Objects stick together. $m_1vec{v}_1 + m_2vec{v}_2 = (m_1+m_2)vec{V}_{final}$. Maximum loss of kinetic energy. * Recoil: Gun-bullet, man-boat. Initial momentum is often zero.

$0 = m_1vec{v}_1 + m_2vec{v}_2$. * Explosions: An object breaking into fragments. Initial momentum equals the vector sum of final momenta of all fragments.

1
Impulse ($vec{J}$): — Change in momentum. $vec{J} = Deltavec{p} = vec{p}_{final} - vec{p}_{initial}$. Also, $vec{J} = int vec{F} dt = vec{F}_{avg}Delta t$. Unit: N·s or kg·m/s.
2
Key Points for Problems:

* Always define the system clearly. * Identify if net external force is zero or negligible. * Treat momentum as a vector; use signs for 1D, components for 2D. * Do not confuse momentum conservation with kinetic energy conservation.

Vyyuha Quick Recall

My Velocity Conserves Momentum:

Mass x Velocity = Momentum ($vec{p} = mvec{v}$)
Conservation of Momentum: If no External Force ($vec{F}_{ext}=0$), then total momentum is Constant ($vec{P}_{sys} = \text{constant}$).
Internal forces Don't Change total momentum.