Reference · 44 terms

Glossary

Short, precise definitions of the words and symbols used in this module, each linked to the lesson where it is taught. Filter the list as you type, jump to a letter, or look a symbol up in Symbols at a glance.

Showing all 44 terms

A

Approach, relative velocity of: How fast two bodies close on each other along the line of impact just before they collide: \({(v_A)_1 - (v_B)_1}\), with one positive direction for both. An impact happens only if it is positive. It is the denominator of the coefficient of restitution.; See: Lesson 6Related: Separation, relative velocity of, Coefficient of restitution \(e\), Line of impact
Area under the force–time graph: The impulse of a force over an interval is the signed area between its \(F\)–\(t\) graph and the time axis: \({I = \int F\,dt}\). Areas below the axis count as negative impulse. Straight-line graphs split into rectangles and triangles.; See: Lesson 2 Lesson 4Related: Impulse \(\Ivec\), Average force, Time-varying force
Average force: The constant force that gives the same impulse over the same time: \({F_\text{avg} = \frac{1}{\Delta t}\int F\,dt}\). In an impact it is the impulse divided by the contact time; stretching the time, with an airbag or a crumple zone, lowers it.; See: Lesson 1 Lesson 2Related: Impulse \(\Ivec\), Contact time, Impulsive force

B

Ballistic pendulum: A heavy block hanging from cords, into which a bullet is fired. The plastic impact conserves horizontal momentum and loses almost all the kinetic energy; the swing that follows conserves energy. The height of the swing gives the bullet's speed: \({v = \frac{m + M}{m}\sqrt{2gh}}\).; See: Lesson 8Related: Perfectly plastic impact, Problems in stages, Conservation of linear momentum

C

Central impact: An impact in which the line of impact passes through the mass centers of both bodies, as for two spheres or two blocks meeting face to face. It is direct when both velocities lie along the line of impact and oblique when they do not.; See: Lesson 6Related: Direct central impact, Oblique impact, Line of impact
Coefficient of restitution \(e\): The ratio of the restitution impulse to the deformation impulse in an impact, \({e = \int R\,dt / \int P\,dt}\), which equals the relative velocity of separation over the relative velocity of approach: \({e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}}\). It runs from \(0\) (perfectly plastic) to \(1\) (perfectly elastic) and depends on the materials, the speed and the shapes.; See: Lesson 6Related: Perfectly elastic impact, Perfectly plastic impact, Drop test
Common velocity (at maximum deformation): At the instant of maximum deformation in an impact, both bodies move with the same velocity \({v = \frac{m_A(v_A)_1 + m_B(v_B)_1}{m_A + m_B}}\), the velocity of their mass center. In a perfectly plastic impact they keep it.; See: Lesson 6Related: Deformation phase, Perfectly plastic impact, Mass center, velocity of the
Components of the principle: The principle of impulse and momentum is a vector equation, so in a plane it gives two scalar equations: \({m(v_x)_1 + \sum\int F_x\,dt = m(v_x)_2}\) and the same for \(y\). It can find two unknowns, such as a velocity and a normal force.; See: Lesson 3Related: Principle of linear impulse and momentum, Linear momentum \(\Lvec\)
Conservation of linear momentum: When the external impulse on a system is zero or negligible in some direction, the total momentum in that direction does not change: \({\sum m_i(\vvec_i)_1 = \sum m_i(\vvec_i)_2}\). It holds direction by direction, and says nothing about the energy.; See: Lesson 5Related: System of particles, Impulsive force, Internal force
Contact time: How long two bodies touch during an impact, often a few milliseconds. The impulse is fixed by the change in momentum, so the shorter the contact, the larger the average force.; See: Lesson 1 Lesson 2Related: Average force, Impulsive force

D

Deformation phase: The first part of an impact, from first contact to maximum deformation, while the contact force \(P\) squeezes the bodies until they move at a common velocity. Its impulse is \(\int P\,dt\).; See: Lesson 6Related: Restitution phase, Common velocity, Coefficient of restitution \(e\)
Direct central impact: A central impact in which both velocities lie along the line of impact: a head-on collision. The velocities afterward come from conservation of momentum and the coefficient of restitution, two equations for two unknowns.; See: Lesson 6Related: Central impact, Coefficient of restitution \(e\), Oblique impact
Drop test: A way to measure \(e\): drop a ball from rest from height \(h_1\) onto a floor and measure the rebound height \(h_2\). Since the floor does not move and each speed is \(\sqrt{2gh}\), \({e = \sqrt{h_2/h_1}}\). Each bounce reaches \(e^2\) times the height of the one before.; See: Lesson 6Related: Coefficient of restitution \(e\), Rebound

E

Energy lost in an impact: The kinetic energy that an impact turns into deformation, heat and sound: \({\Delta T = \frac{m_A m_B}{2(m_A + m_B)}\left(1 - e^2\right)\left[(v_A)_1 - (v_B)_1\right]^2}\). It is zero only when \(e = 1\) and largest when \(e = 0\).; See: Lesson 6Related: Coefficient of restitution \(e\), Perfectly plastic impact, Kinetic energy \(T\)
External force: A force on a body of a system from something outside the system, such as the floor, a wall, a support or the Earth's gravity. Only external impulses change the total momentum of a system.; See: Lesson 5Related: Internal force, System of particles, Conservation of linear momentum

I

Impact: A collision between two bodies in which very large forces act for a very short time. During it, positions hardly change while velocities change abruptly, so it is treated as instantaneous.; See: Lesson 6Related: Impulsive force, Central impact, Coefficient of restitution \(e\)
Impulse \(\Ivec\) (linear): The integral of a force over the time it acts: \({\Ivec = \int_{t_1}^{t_2}\Fvec\,dt}\), a vector in \(\text{N}{\cdot}\text{s}\). For a constant force it is \({\Fvec\,\Delta t}\). Every force acting during an interval has an impulse, even one that does no work.; See: Lesson 2Related: Linear momentum \(\Lvec\), Area under the force–time graph, Principle of linear impulse and momentum
Impulse and momentum diagrams: Three drawings of a particle: its momentum at \(t_1\), the impulses of all the forces over the interval, and its momentum at \(t_2\). The vectors of the first two add up to the third. The middle drawing is the free-body diagram with each force multiplied by its time.; See: Lesson 3Related: Principle of linear impulse and momentum, Impulse \(\Ivec\)
Impulsive force: A force that is very large and acts for a very short time, so that its impulse is significant even though the time is not: the contact forces in a collision, the push of a rigid floor or wall that is struck, the pull of a cord that jerks taut.; See: Lesson 2 Lesson 5Related: Nonimpulsive force, Impact, Average force
Inertial frame: A frame of reference that does not accelerate, in which Newton's laws hold; for these problems, the ground. Velocities in the principle of impulse and momentum must be measured in it, so a speed given relative to a moving body must be converted first.; See: Lesson 3 Lesson 5Related: Relative velocity, Principle of linear impulse and momentum
Internal force: A force between two bodies of the same system. By Newton's third law internal forces come in equal, opposite, collinear pairs acting for the same time, so their impulses cancel and they cannot change the system's total momentum.; See: Lesson 5Related: External force, Newton's third law, Conservation of linear momentum

K

Kinetic energy \(T\): \({T = \tfrac12 mv^2}\), a scalar. Unlike momentum it does not depend on direction, and it is not conserved in an impact unless \(e = 1\). For a given momentum \(mv\), \({T = (mv)^2/2m}\): a light body carries more of it.; See: Lesson 2 Lesson 6Related: Linear momentum \(\Lvec\), Energy lost in an impact

L

Line of impact: The common normal to the surfaces of two bodies at their point of contact. The impulsive contact force acts along it on smooth surfaces, and the coefficient of restitution applies to the velocity components along it.; See: Lesson 6 Lesson 7Related: Normal axis \(n\), Central impact, Oblique impact
Linear momentum \(\Lvec\): Mass times velocity, \({\Lvec = m\vvec}\): a vector in the direction of the velocity, in \(\text{kg}{\cdot}\text{m/s}\). Beer & Johnston write it \(m\vvec\) and Meriam & Kraige \(\mathbf G\). A body's momentum changes only through the impulse of the forces on it.; See: Lesson 2Related: Impulse \(\Ivec\), Principle of linear impulse and momentum, Kinetic energy \(T\)

M

Mass center, velocity of the: The total momentum of a system equals its total mass times the velocity of its mass center: \({\sum m_i\vvec_i = m\vvec_G}\). External impulses change it; internal forces never can.; See: Lesson 5Related: System of particles, Common velocity, Internal force

N

Newton's cradle: A row of hanging steel balls. When one strikes the row, the impulse passes along it and the last ball swings out: equal masses in a nearly elastic impact exchange velocities.; See: Lesson 6Related: Perfectly elastic impact, Direct central impact
Newton's third law: The forces two bodies exert on each other are equal in size, opposite in direction and collinear. Because they also act for the same time, their impulses are equal and opposite, which is why internal forces drop out of a system's momentum.; See: Lesson 5Related: Internal force, Conservation of linear momentum
Nonimpulsive force: A force that stays finite, such as a weight, a spring force or ordinary friction. Over the very short time of an impact its impulse is negligible, so it can be left out of the momentum balance for the impact itself.; See: Lesson 5Related: Impulsive force, Conservation of linear momentum, Problems in stages
Normal axis \(n\): In an oblique impact, the axis along the line of impact. Along \(n\), the momentum of the pair is conserved and the coefficient of restitution applies, exactly as in a direct impact.; See: Lesson 7Related: Tangential axis \(t\), Line of impact, Oblique impact

O

Oblique impact: An impact in which one or both velocities are not along the line of impact. With smooth surfaces, each body keeps its own tangential component, and the normal components behave as in a direct impact.; See: Lesson 7Related: Normal axis \(n\), Tangential axis \(t\), Direct central impact

P

Perfectly elastic impact: An impact with \(e = 1\): the bodies separate as fast as they approached, and no kinetic energy is lost. It is an ideal; hard steel and glass come close.; See: Lesson 6Related: Perfectly plastic impact, Coefficient of restitution \(e\), Newton's cradle
Perfectly plastic impact: An impact with \(e = 0\): there is no restitution, so the bodies move on together at their common velocity, as when railcars couple or a bullet embeds in a block. The kinetic energy lost is the largest possible; momentum is still conserved.; See: Lesson 5 Lesson 6Related: Perfectly elastic impact, Common velocity, Energy lost in an impact
Principle of linear impulse and momentum: A particle's momentum at \(t_1\), plus the impulses of all the forces on it from \(t_1\) to \(t_2\), equals its momentum at \(t_2\): \({m\vvec_1 + \sum\int_{t_1}^{t_2}\Fvec\,dt = m\vvec_2}\). It is Newton's second law integrated in time, and relates velocity to time.; See: Lesson 1 Lesson 3Related: Impulse \(\Ivec\), Linear momentum \(\Lvec\), Components of the principle
Problems in stages: Problems in which the physics changes during the motion, such as a slide, a collision and a skid. Cut the motion at each impact and other change; use conservation of momentum and \(e\) across each impact, energy or impulse on either side, and carry the velocity across each cut.; See: Lesson 8Related: Ballistic pendulum, Nonimpulsive force, Energy lost in an impact

R

Rebound: Motion back the way a body came after an impact. Its velocity has the opposite sign, so the change in velocity is the sum of the two speeds: a ball arriving at \(40\ \text{m/s}\) and leaving at \(50\ \text{m/s}\) changes by \(90\ \text{m/s}\).; See: Lesson 2 Lesson 6Related: Drop test, Coefficient of restitution \(e\)
Recoil: The backward motion of a gun, a cannon or a person when a projectile or a jump carries momentum forward. With no external impulse in that direction, the momentum of the recoiling body is equal and opposite to that of the projectile.; See: Lesson 5Related: Conservation of linear momentum, Internal force
Relative velocity: The velocity of one body as seen from another that moves: \({\vvec_{A/B} = \vvec_A - \vvec_B}\). Momentum must be written with velocities relative to the ground, so a speed given relative to a cart or boat is converted first.; See: Lesson 5Related: Inertial frame, Approach, relative velocity of, Separation, relative velocity of
Restitution phase: The second part of an impact, from maximum deformation until the bodies separate, while the contact force \(R\) pushes them apart as they recover their shape, fully or partly. Its impulse is \(\int R\,dt = e\int P\,dt\).; See: Lesson 6Related: Deformation phase, Coefficient of restitution \(e\)

S

Separation, relative velocity of: How fast two bodies move apart along the line of impact just after they collide: \({(v_B)_2 - (v_A)_2}\). It is \(e\) times the relative velocity of approach.; See: Lesson 6Related: Approach, relative velocity of, Coefficient of restitution \(e\)
Smooth surface: A surface that exerts no friction. In an oblique impact between smooth bodies the contact force is purely normal, so it gives no tangential impulse, and each body keeps its tangential velocity.; See: Lesson 7Related: Tangential axis \(t\), Oblique impact
Starting to slide (static friction): A body at rest on a rough surface stays put while the push is less than \(\mu_s N\): static friction matches the push and their impulses cancel. It starts to slide when the push reaches \(\mu_s N\), and the impulse calculation starts at that instant.; See: Lesson 4Related: Time-varying force, Impulse \(\Ivec\)
System of particles: Any set of particles treated together. Its total momentum changes only through external impulses: \({\sum m_i(\vvec_i)_1 + \sum\int\Fvec_{\text{ext}}\,dt = \sum m_i(\vvec_i)_2}\).; See: Lesson 5Related: Internal force, External force, Conservation of linear momentum

T

Tangential axis \(t\): In an oblique impact, the axis along the common tangent to the contact surfaces. On smooth surfaces there is no impulse along it, so each body's tangential velocity component is the same before and after.; See: Lesson 7Related: Normal axis \(n\), Smooth surface, Oblique impact
Time-varying force: A force given as a function of time, such as \(kt\) or \(F_0\sin(\pi t/T)\). Its impulse is \(\int F(t)\,dt\), which makes impulse and momentum the natural method. A force that depends on position calls for work and energy instead.; See: Lesson 4Related: Impulse \(\Ivec\), Area under the force–time graph, Starting to slide

Symbols at a glance

Met a symbol in a lesson or in another textbook and not sure what it stands for? Find it here, then follow the link to its entry.

Symbols used in this module and in other books
Symbol	Meaning	Entry
\(\colL{\Lvec = m\vvec}\)	Linear momentum	Linear momentum \(\Lvec\)
\(\Ivec = \int\Fvec\,dt\)	Linear impulse	Impulse \(\Ivec\)
\(F_\text{avg}\)	Average force over an interval	Average force
\((v_A)_1,\ (v_A)_2\)	Velocity of \(A\) just before and just after an impact	Direct central impact
\(v\)	Common velocity at maximum deformation	Common velocity
\(e\)	Coefficient of restitution	Coefficient of restitution \(e\)
\(\int P\,dt,\ \int R\,dt\)	Deformation and restitution impulses	Deformation phase, Restitution phase
\(\Delta T\)	Kinetic energy lost in an impact	Energy lost in an impact
\(n,\ t\)	Normal and tangential axes at the contact	Normal axis \(n\), Tangential axis \(t\)
\(\mu_s,\ \mu_k\)	Coefficients of static and kinetic friction	Starting to slide
\(\vvec_G\)	Velocity of the mass center of a system	Mass center, velocity of the