Forces, density and pressure

Linear Momentum and Its Conservation

1. Linear Momentum

Linear momentum is a vector quantity that describes the motion of an object. It is defined as the product of an object's mass and its velocity:

\$\mathbf{p}=m\mathbf{v}\$

• Magnitude: \$p=mv\$
• Direction: same as the velocity vector
• Units: kilogram metre per second (kg·m/s)

2. Conservation of Linear Momentum

In an isolated system (no external forces), the total linear momentum before an interaction equals the total linear momentum after the interaction:

\$\sum \mathbf{p}{\text{initial}}=\sum \mathbf{p}{\text{final}}\$

1. Elastic collision – kinetic energy is also conserved.
2. Inelastic collision – kinetic energy is not conserved, but momentum is.
3. Impulse–momentum theorem – the change in momentum equals the impulse applied:

\$\mathbf{J}=\Delta \mathbf{p}=\int \mathbf{F}\,dt\$

• Impulse is the area under a force–time graph.
• For a constant force over a time interval \$\Delta t\$, \$J=F\Delta t\$.

3. Forces, Density and Pressure

Force is related to the rate of change of momentum:

\$\mathbf{F}=\frac{d\mathbf{p}}{dt}\$

In fluid mechanics, density and pressure play a key role in momentum transfer:

• Density – mass per unit volume:

\$\rho=\frac{m}{V}\$

• Pressure – force per unit area:

\$p=\frac{F}{A}\$

• Hydrostatic pressure – pressure at a depth \$h\$ in a fluid of density \$\rho\$:

\$p=p_0+\rho g h\$

• Momentum flux in a fluid – pressure exerted by a moving fluid can be expressed as:

\$p_{\text{dynamic}}=\frac{1}{2}\rho v^2\$

These relations are essential when analysing forces on submerged bodies, fluid jets, and gas flows.

4. Applications and Examples

Example 1: Two blocks of masses \$m1\$ and \$m2\$ collide on a frictionless surface. If block 1 moves with velocity \$v_1\$ and block 2 is initially at rest, the final velocities after an elastic collision are:

\$v1'=\frac{m1-m2}{m1+m2}v1,\qquad v2'=\frac{2m1}{m1+m2}v_1\$

Example 2: A water jet of density \$\rho\$ and velocity \$v\$ strikes a flat plate. The force exerted on the plate is the rate of change of momentum:

\$F=\dot{m}v=\rho A v^2\$

where \$A\$ is the cross‑sectional area of the jet.

5. Summary Table