Define force as the rate of change of momentum and, using this definition together with Newton’s three laws, solve quantitative problems involving constant‑force motion, variable‑mass systems, elastic and inelastic collisions, and non‑uniform motion (friction, air‑resistance and terminal velocity).
\[
\mathbf{F}_{\text{net}}=\frac{d\mathbf{p}}{dt}.
\]
If the mass does not change (\(dm/dt=0\)):
\[
\mathbf{F}= \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt}=m\mathbf{a}.
\]
When the mass varies with time:
\[
\mathbf{F}= \frac{d}{dt}(m\mathbf{v}) = m\frac{d\mathbf{v}}{dt}+ \mathbf{v}\frac{dm}{dt}.
\]
The extra term \(\mathbf{v}\,dm/dt\) accounts for the momentum carried away (or added) with the ejected (or accreted) mass.
\[
\int{t1}^{t2}\mathbf{F}\,dt = \mathbf{p}(t2)-\mathbf{p}(t_1)=\Delta\mathbf{p}.
\]
If a constant force \(\mathbf{F}\) acts for a duration \(\Delta t\):
\[
\Delta\mathbf{p}= \mathbf{F}\,\Delta t,\qquad
\mathbf{v}{f}= \mathbf{v}{i}+ \frac{\mathbf{F}}{m}\,\Delta t.
\]
Rocket of instantaneous mass \(m\) ejects exhaust at speed \(\mathbf{u}\) relative to the rocket (opposite to \(\mathbf{v}\)). The thrust is
\[
\mathbf{F}_{\text{thrust}} = -\mathbf{u}\,\frac{dm}{dt}.
\]
Including external forces (gravity, drag):
\[
m\frac{d\mathbf{v}}{dt}= -\mathbf{u}\frac{dm}{dt} + \mathbf{F}_{\text{ext}}.
\]
For a closed system (\(\sum\mathbf{F}_{\text{ext}}=0\)):
\[
\frac{d}{dt}\Bigl(\sum\mathbf{p}\Bigr)=0
\;\Longrightarrow\;
\sum\mathbf{p}{\text{initial}}=\sum\mathbf{p}{\text{final}}.
\]
This principle underlies all collision problems.
In many real situations the net force is not constant because it contains a resistive component that depends on speed. Typical forms are:
When a falling object reaches a speed at which the downward weight \(mg\) is exactly balanced by the upward resistive force, the net force becomes zero and the object moves at a constant speed – the terminal velocity \(vt\). Using \(F{\text{net}}=0\) gives, for linear drag, \(mg=bvt\) and for quadratic drag, \(mg=cvt^{2}\). The same momentum‑force relation (\(\mathbf{F}=d\mathbf{p}/dt\)) applies; the resistive force simply appears as part of \(\mathbf{F}_{\text{net}}\).
Problem: A 0.150 kg ball moving at \(8.0\ \text{m s}^{-1}\) collides head‑on with a 0.250 kg ball initially at rest. The impact lasts \(0.020\ \text{s}\) and the average force on each ball is \(120\ \text{N}\). Find the final velocities assuming a perfectly elastic collision.
\[
v_{1f}= -2.0\ \text{m s}^{-1},\qquad
v_{2f}= 5.6\ \text{m s}^{-1}.
\]
Problem: A 1.20 kg cart moving at \(3.0\ \text{m s}^{-1}\) collides with a stationary 0.80 kg cart. The carts lock together after the impact, which lasts \(0.050\ \text{s}\). The average force during the collision is \(40\ \text{N}\). Find the common final speed.
\[
vf = \frac{pf}{m_{\text{tot}}}= \frac{3.6}{2.0}=1.8\ \text{m s}^{-1}.
\]
\[
K_i=\tfrac12(1.20)(3.0)^2=5.4\ \text{J},\qquad
K_f=\tfrac12(2.00)(1.8)^2=3.24\ \text{J}.
\]
The loss (≈2.2 J) appears as deformation, heat, sound, etc.
Problem: A rocket of mass \(m=500\ \text{kg}\) ejects propellant at a relative speed \(u=2500\ \text{m s}^{-1}\) and a mass‑flow rate \(|dm/dt| = 5\ \text{kg s}^{-1}\). Neglect gravity and drag. Find the thrust and the initial acceleration.
\[
F_{\text{thrust}} = u\left|\frac{dm}{dt}\right| = 2500\times5 = 1.25\times10^{4}\ \text{N}.
\]
\[
a = \frac{F_{\text{thrust}}}{m}= \frac{1.25\times10^{4}}{500}=25\ \text{m s}^{-2}.
\]
| Quantity | Symbol | Definition / Equation | SI Unit |
|---|---|---|---|
| Momentum | \(\mathbf{p}\) | \(\mathbf{p}=m\mathbf{v}\) | kg·m·s⁻¹ |
| Force (general) | \(\mathbf{F}\) | \(\mathbf{F}=d\mathbf{p}/dt\) | N (kg·m·s⁻²) |
| Constant‑mass form | \(\mathbf{F}=m\mathbf{a}\) | ||
| Variable‑mass form | \(\mathbf{F}=m\mathbf{a}+\mathbf{v}\,dm/dt\) | ||
| Impulse | \(\mathbf{J}\) | \(\mathbf{J}= \int\mathbf{F}\,dt = \Delta\mathbf{p}\) | kg·m·s⁻¹ |
| Conservation of linear momentum | \(\displaystyle\sum\mathbf{p}{\text{initial}}=\sum\mathbf{p}{\text{final}}\) (closed system) | ||
| Linear friction | \(\mathbf{F}_{\text{fr}}\) | \(-\mu_k N\,\hat{\mathbf{v}}\) | N |
| Linear drag | \(\mathbf{F}_{\text{drag}}\) | \(-b\,\mathbf{v}\) | N |
| Quadratic drag | \(\mathbf{F}_{\text{drag}}\) | \(-c\,v^{2}\,\hat{\mathbf{v}}\) | N |
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