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 $m_1$ and $m_2$ 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:

$$v_1'=\frac{m_1-m_2}{m_1+m_2}v_1,\qquad v_2'=\frac{2m_1}{m_1+m_2}v_1$$

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