ReferenceImpulse and Momentum
Formula Sheet
Every key result from the module in one place. Conventions: Hibbeler notation (\(\Lvec = m\vvec\), \(\int\Fvec\,dt\), \((v_A)_1\) before and \((v_A)_2\) after an impact, \(e\)); one positive direction for each axis, with signed velocities; SI units with \(g = 9.81\ \text{m/s}^2\).
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Momentum and impulse
Linear momentum
\[ \colL{\Lvec = m\vvec} \]Linear impulse
\[ \Ivec = \int_{t_1}^{t_2}\Fvec\,dt \]Both are vectors; \(1\ \text{N}{\cdot}\text{s} = 1\ \text{kg}{\cdot}\text{m/s}\). A constant force: \(\Ivec = \Fvec\,\Delta t\). A force that varies: the signed area under its \(F\)–\(t\) graph. Average force: \(F_\text{avg} = I/\Delta t\).
Every force has an impulse over the time it acts, including the weight and normal forces that do no work.
More in Lesson 2
The principle of impulse and momentum
In components: \(m(v_x)_1 + \sum\int F_x\,dt = m(v_x)_2\), and the same for \(y\).
Procedure: (1) interval and positive directions; (2) free-body diagram, then the impulse and momentum diagrams; (3) each impulse, \(F\,\Delta t\) or \(\int F\,dt\); (4) one equation per direction; solve and check signs.
- Find normal forces from the direction in which nothing moves.
- If friction reverses, split the interval where \(v = 0\).
More in Lesson 3
Forces that vary with time
| Force | \(\int_0^t F\,dt\) |
|---|---|
| \(kt\) | \(\tfrac12 kt^2\) |
| \(ct^2\) | \(\tfrac13 ct^3\) |
| \(F_0\sin(\pi t/T)\), \(0 \le t \le T\) | \(2F_0T/\pi\) over the whole pulse |
- Static friction: a body at rest starts to slide when the push reaches \(\mu_s N\). Start the interval there.
- Largest speed where the net force is zero.
- Force given as a function of position: use work and energy instead.
More in Lesson 4
Systems and conservation of momentum
Internal forces cancel in pairs. If the external impulse in a direction is zero or negligible, \(\sum m_i(\vvec_i)_1 = \sum m_i(\vvec_i)_2\) in that direction.
Nonimpulsive (neglect in a short impact): weights, springs, ordinary friction. Impulsive (keep): contact forces, rigid floors and walls, taut cords.
Convert speeds given relative to a moving body to ground velocities first.
More in Lesson 5
Direct central impact
One positive direction for all four velocities. \(e\): separation over approach, \(0 \le e \le 1\). At maximum deformation both bodies move at \(v = \dfrac{m_A(v_A)_1 + m_B(v_B)_1}{m_A + m_B}\).
More in Lesson 6
Restitution and energy
| \(e\) | Impact | Kinetic energy |
|---|---|---|
| \(1\) | perfectly elastic | conserved |
| \(0 \lt e \lt 1\) | partly elastic | partly lost |
| \(0\) | perfectly plastic: move together | largest loss |
Drop test on a fixed floor: \(e = \sqrt{h_2/h_1}\); each bounce reaches \(e^2\) times the height before.
More in Lesson 6
Oblique impact
\(n\) along the line of impact, \(t\) along the common tangent. On smooth surfaces:
- Along \(t\): each body keeps its own component, \((v_t)_2 = (v_t)_1\).
- Along \(n\): momentum of the pair and \(e\), as in direct impact.
Ball on a fixed smooth surface (angles from the surface)
\[ (v_n)_2 = e\,(v_n)_1 \qquad \tan\theta_2 = e\tan\theta_1 \]Equal masses, one at rest, \(e = 1\): they move off at \(90^\circ\).
More in Lesson 7
Choosing a method
| You need | Use |
|---|---|
| velocity after a time; force over time | impulse and momentum |
| velocities just after a collision | momentum and \(e\) |
| speed at a position | work and energy |
| acceleration or force at an instant | \(\sum\Fvec = m\avec\) |
In stages: cut at each impact; momentum and \(e\) across it, energy (or impulse) on either side; carry the velocity across each cut.
More in Lesson 8
Common mistakes
- \(50 - 40\) for a rebound. Velocities have signs: \(50 - (-40)\).
- "No work, so no impulse." The normal force has an impulse.
- Starting the impulse at \(t = 0\) while static friction still holds the body.
- Conserving momentum with an impulsive floor, wall or cord acting.
- Conserving energy across an impact. Only when \(e = 1\).
- \(e\) upside down, or applied to the whole velocity in an oblique impact.
- Milliseconds not converted before dividing by a contact time.
A solution, step by step
Block \(A\) (\(2\ \text{kg}\), \(6\ \text{m/s}\)) strikes block \(B\) (\(3\ \text{kg}\)) at rest against a spring, \(k = 600\ \text{N/m}\), with \(e = 0.5\). Find the maximum compression.
- Cut at the impact; the spring is nonimpulsive during it.
- Momentum: \(12 = 2(v_A)_2 + 3(v_B)_2\).
- Restitution: \((v_B)_2 - (v_A)_2 = 0.5(6) = 3\), so \((v_B)_2 = 3.6\ \text{m/s}\), \((v_A)_2 = 0.6\ \text{m/s}\).
- Energy for \(B\): \(\tfrac12(3)(3.6)^2 = \tfrac12(600)s^2\), so \(s = 0.255\ \text{m}\).
- Check: \(16.2\ \text{J}\) of \(36\ \text{J}\) was lost in the impact; units in J and m.
The full solution is Example 8.1 in Lesson 8
Check values
Test your method or calculator on these (\(g = 9.81\ \text{m/s}^2\)).
| \(0.145\ \text{kg}\) ball, \(40 \to -50\ \text{m/s}\), \(1.2\ \text{ms}\) | \(F_\text{avg} = 10.9\ \text{kN}\) |
|---|---|
| \(20\ \text{kg}\), \(P = 30t\), \(\mu_s = 0.4\), \(\mu_k = 0.3\) | slides at \(2.616\ \text{s}\); \(v(6) = 8.91\ \text{m/s}\) |
| \(20\ \text{Mg}\) at \(1.5\ \text{m/s}\) couples with \(15\ \text{Mg}\) | \(0.857\ \text{m/s}\); \(9.64\ \text{kJ}\) lost |
| \(1\ \text{kg}\) at \(5\), \(2\ \text{kg}\) at \(-2\ \text{m/s}\), \(e = 0.6\) | \(-2.47\), \(1.73\ \text{m/s}\) |
| \(12\ \text{m/s}\) at \(50^\circ\) on a smooth floor, \(e = 0.6\) | \(9.48\ \text{m/s}\) at \(35.6^\circ\) |
| \(20\ \text{g}\) at \(600\ \text{m/s}\) into \(4\ \text{kg}\) | \(2.99\ \text{m/s}\); rises \(0.454\ \text{m}\) |
Symbols and units
| Symbol | Meaning | Unit |
|---|---|---|
| \(\colL{\Lvec = m\vvec}\) | linear momentum | kg·m/s |
| \(\Ivec = \int\Fvec\,dt\) | linear impulse | N·s |
| \((v_A)_1,\ (v_A)_2\) | velocity of \(A\) just before, just after | m/s |
| \(e\) | coefficient of restitution | none |
| \(\int P\,dt,\ \int R\,dt\) | deformation, restitution impulse | N·s |
| \(n,\ t\) | normal, tangential axes at contact | none |
| \(\mu_s,\ \mu_k\) | static, kinetic friction coefficient | none |
Put the formulas to work in the Practice Lab, test yourself with the Self-Check Quiz, or look up a term in the Glossary.