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1. KinematicsSavchenko Formulas, chapter 1 of 14, 31 formulas

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1.1Constant-speed motion

Uniform motion Равномерное движение

law 1.1

$$x = x_0 + vt$$

\displaystyle s = vt

\displaystyle v = \frac{s}{t}

x: coordinate at time $$t$$
x₀: initial coordinate
v: velocity
s: distance covered

At constant velocity the distance grows in proportion to time and the coordinate changes linearly. The velocity is the distance over the time, and the same formula gives the travel time of a signal or of light over a given distance.

Average speed Средняя скорость

definition 1.1

ср$$v_{\text{ср}} = \frac{S}{t}$$

ср$\displaystyle v_{\text{ср}} = \frac{2v_1 v_2}{v_1 + v_2}$

S: the whole distance covered
t: the whole time of motion
v₁, v₂: speeds on the two halves of the path

Average speed is the whole distance over the whole time, stops included. When two halves of a path are covered at $$v_1$$ and $$v_2$$ , the average speed is their harmonic mean, not the arithmetic one.

Appears in problems (10) 1.1.15 1.2.1 1.2.7 1.3.1 2.3.29 3.3.36 5.4.8 5.4.16

Closing and separating speed Скорость сближения и удаления

method 1.1

t = \frac{l}{v_1 + v_2}

сбл$\displaystyle v_{\text{сбл}} = v_1 + v_2$

уд$\displaystyle v_{\text{уд}} = |v_1 - v_2|$

l: distance between the bodies
v₁, v₂: velocities of the bodies

Bodies moving towards each other close at $$v_1 + v_2$$ , bodies moving the same way catch up at $$|v_1 - v_2|$$ . The time to meet or to overtake is the distance over that speed.

Appears in problems (5) 1.1.6 1.2.5 1.4.7 2.7.45 14.1.6

Velocity from its components Скорость по компонентам

definition 1.1

v = \sqrt{v_x^2 + v_y^2 + v_z^2}

\displaystyle v_x = v\cos\alpha,\quad v_y = v\sin\alpha

v_x, v_y, v_z: components of the velocity along the axes
$\alpha$: angle between the velocity and the $$x$$ axis

Velocity is a vector. Its magnitude follows from the components by Pythagoras, the components from the magnitude and the angle. Velocities add and subtract component by component or by the triangle rule.

Appears in problems (3) 1.1.3 1.4.9 14.4.4

1.2Variable-speed motion

Uniformly accelerated motion Равноускоренное движение

law 1.2

x = x_0 + v_0 t + \frac{at^2}{2}, \qquad v = v_0 + at

\displaystyle v^2 - v_0^2 = 2a(x - x_0)

\displaystyle x - x_0 = \frac{v_0 + v}{2}\,t

изсостоянияпокоя$\displaystyle v = \sqrt{2as}\ \text{(из состояния покоя)}$

x₀, v₀: coordinate and velocity at the start
a: constant acceleration
s: distance from rest

Under constant acceleration the velocity changes linearly and the coordinate quadratically. The third formula eliminates time and serves when the time is not asked for. Everything here is a signed projection on one axis.

Velocity as the derivative of the coordinate Скорость как производная координаты

definition 1.2

v = \frac{dx}{dt}

\displaystyle \vec v = \frac{d\vec r}{dt}

\displaystyle v = \lim_{\Delta t \to 0}\frac{\Delta x}{\Delta t}

x: coordinate
$\vec r$: position vector
t: time

Instantaneous velocity is the limit of displacement over the time interval, the time derivative of the coordinate. On an $$x(t)$$ graph it is the slope of the tangent. The velocity points along the trajectory.

Acceleration as the derivative of velocity Ускорение как производная скорости

definition 1.2

a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}

\displaystyle \vec a = \frac{d\vec v}{dt}

\displaystyle \Delta v = a\,\Delta t

a: acceleration
v: velocity

Acceleration is the rate of change of the velocity vector, its time derivative. It points where the velocity changes, and along a curved path it does not point along the velocity.

Appears in problems (11) 1.2.16 1.2.17 2.1.49 2.2.41 2.6.42 3.2.32 3.3.29 4.3.18

Distance as the integral of velocity Путь как интеграл скорости

law 1.2

x(t) = x_0 + \int_0^t v\,dt

\displaystyle v(t) = v_0 + \int_0^t a\,dt

\displaystyle \Delta x = v\,\Delta t

$$v(t)$$: velocity as a function of time
$\Delta x$: displacement in a short time $\Delta t$

Displacement is the area under the velocity graph, the change of velocity the area under the acceleration graph. In a short time the path is $v\,\Delta t$ , and adding such pieces gives the integral. This is how problems with a velocity given by a graph or a formula are solved.

Appears in problems (10) 1.2.18 1.5.17 2.2.22 2.2.39 2.3.29 7.3.6 7.3.11 8.1.16

Volume flow rate Объёмный расход

definition 1.2

q = \frac{dV}{dt} = Sv

q: flow rate, volume per unit time
V: volume
S: cross-section area
v: flow velocity

The volume passing a cross-section in a time $$dt$$ is $Sv\,dt$ . When a volume grows at a known rate, the speed of its boundary is the rate over the cross-section area there. The same rate is kept along a pipe.

Appears in problems (6) 1.2.10 1.2.11 1.2.12 4.4.6 4.4.7 8.4.15

Meeting condition Условие встречи

method 1.2

$$x_1(t) = x_2(t)$$

$$x_1(t), x_2(t)$$: the two bodies' coordinates as functions of time on one axis

Two bodies meet when their coordinates coincide. Writing each body's law of motion in one frame and equating the coordinates gives an equation for the meeting time.

Appears in problems (4) 1.1.14 1.2.2 3.4.18 7.1.25

1.3Motion in gravity field. Curvilinear motion

Centripetal acceleration Центростремительное ускорение

law 1.3

a_n = \frac{v^2}{R} = \omega^2 R

\displaystyle a_n = \frac{4\pi^2 R}{T^2}

\displaystyle F_n = \frac{mv^2}{R}

v: speed of the point
R: radius of the circle or radius of curvature
$\omega$: angular velocity
T: period of revolution

On a circle at constant speed the acceleration points to the centre and equals $$v^2/R$$ . For any curve this is the normal component of the acceleration with $$R$$ the radius of curvature at that point. Times the mass it is the net force along the normal.

Angular velocity and period Угловая скорость и период

definition 1.3

\omega = \frac{d\varphi}{dt} = \frac{2\pi}{T} = 2\pi\nu, \qquad v = \omega R

\displaystyle T = \frac{2\pi R}{v}

\displaystyle \nu = \frac{1}{T}

$\varphi$: angle turned
T: period
$\nu$: frequency, revolutions per second
R: radius

Angular velocity is the angle turned per unit time. In one period the body turns by $2\pi$ , and a point at a distance $$R$$ from the axis moves at $\omega R$ . All points of a rotating body share one angular velocity while their speeds grow with the distance to the axis.

Free fall Свободное падение

law 1.3

h = \frac{gt^2}{2}, \qquad v = gt, \qquad v = \sqrt{2gh}

\displaystyle t = \sqrt{\frac{2h}{g}}

\displaystyle v = v_0 - gt,\qquad h = v_0 t - \frac{gt^2}{2}

h: height covered
g: acceleration of free fall, $$9.8$$ m/s $$^2$$
v₀: initial vertical velocity

Near the Earth every body falls with the same downward acceleration $$g$$ , whatever its mass. It is uniformly accelerated motion with $$a = g$$ . A body thrown upward rises for a time $$v_0/g$$ to a height $$v_0^2/2g$$ and returns with the same speed.

Appears in problems (18) 1.3.1 1.3.2 1.3.4 1.4.11 1.4.12 1.5.4 2.1.9 2.2.32

Projectile motion Тело, брошенное под углом к горизонту

law 1.3

x = v_0 t\cos\alpha, \qquad y = v_0 t\sin\alpha - \frac{gt^2}{2}

v₀: initial speed
$\alpha$: launch angle above the horizontal
x, y: horizontal and vertical coordinates

The horizontal motion is uniform, the vertical one uniformly accelerated with $$g$$ downward, and the two are independent. Writing both coordinates as functions of time, one eliminates the time or finds it from one equation and substitutes into the other.

Appears in problems (9) 1.3.6 1.3.10 1.3.11 1.3.13 1.3.15 1.3.18 1.3.27 1.3.30

Tangential and normal acceleration Тангенциальное и нормальное ускорения

law 1.3

a_\tau = \frac{dv}{dt}, \qquad a_n = \frac{v^2}{R}, \qquad a = \sqrt{a_\tau^2 + a_n^2}

$a_\tau$: component along the velocity, changes the speed
a_n: normal component, turns the velocity
R: radius of curvature

Acceleration splits into a tangential part that changes the speed and a normal part that turns the velocity. The total acceleration is their vector sum. In uniform motion along a curve only the normal part remains.

Appears in problems (9) 1.3.24 1.3.25 1.3.29 1.5.12 2.1.53 2.1.63 2.1.64 2.3.22

Velocity components of a projectile Скорость брошенного тела по компонентам

law 1.3

v_x = v_0\cos\alpha, \qquad v_y = v_0\sin\alpha - gt

v_x: horizontal component, unchanged
v_y: vertical component
$\alpha$: launch angle

The horizontal velocity component is conserved, the vertical one drops by $$g$$ every second. At the top $$v_y = 0$$ . The angle of the velocity to the horizontal at any moment is given by $\tan\varphi = v_y/v_x$ .

Appears in problems (8) 1.3.5 1.3.6 1.3.14 1.3.27 1.3.30 1.4.12 2.4.16 7.3.9

Maximum height and time of flight Высота и время полёта

law 1.3

H = \frac{v_0^2\sin^2\alpha}{2g}, \qquad T = \frac{2v_0\sin\alpha}{g}

под$\displaystyle t_{\text{под}} = \frac{v_0\sin\alpha}{g}$

нанаклоннойплоскостиподуглом$\displaystyle T = \frac{2v_0\sin\alpha}{g\cos\beta}\ \text{(на наклонной плоскости под углом }\beta)$

H: maximum height
T: time of flight back to the launch height
$под$t_{\text{под}}$$: time to the highest point

At the top the vertical velocity vanishes, which gives the rise time $v_0\sin\alpha/g$ and the height $v_0^2\sin^2\alpha/2g$ . The descent takes as long as the rise, so the total time is twice that. On an incline the axes are best taken along it and normal to it.

Appears in problems (7) 1.3.6 1.3.10 1.3.12 1.3.16 1.3.27 1.4.11 2.3.19

Acceleration along a smooth incline Ускорение вдоль гладкой наклонной плоскости

law 1.3

a = g\sin\alpha

вдольхордыподугломквертикали$\displaystyle a = g\cos\varphi\ \text{(вдоль хорды под углом }\varphi\text{ к вертикали)}$

$\alpha$: angle of the incline to the horizontal
$g\cos\alpha$: component of $$g$$ normal to the plane

Along a direction at an angle to the vertical only the projection of $$g$$ on that direction acts. A body slides down a smooth incline with $g\sin\alpha$ and along a chord at an angle $\varphi$ to the vertical with $g\cos\varphi$ .

Appears in problems (7) 1.3.2 1.3.3 1.3.28 1.5.4 2.1.49 2.3.22 3.2.6

Range of a projectile Дальность полёта

law 1.3

L = \frac{v_0^2\sin 2\alpha}{g}

\displaystyle L = \frac{2v_0^2\sin\alpha\cos\alpha}{g}

при$\displaystyle L_{\max} = \frac{v_0^2}{g}\ \text{при}\ \alpha = 45^\circ$

L: horizontal range
v₀: initial speed
$\alpha$: launch angle

The flight time back to the same height is $2v_0\sin\alpha/g$ , and times the horizontal speed $v_0\cos\alpha$ it gives the range. The range is largest at $45^\circ$ , and the angles $\alpha$ and $90^\circ - \alpha$ give the same range.

Appears in problems (5) 1.3.6 1.3.12 1.3.18 1.3.19 1.3.27

Trajectory equation Уравнение траектории

law 1.3

y = x\tan\alpha - \frac{g x^2}{2 v_0^2\cos^2\alpha}

\displaystyle y = x\tan\alpha - \frac{g x^2}{2v_0^2}\,(1 + \tan^2\alpha)

x, y: coordinates of a point of the path
v₀: initial speed
$\alpha$: launch angle

Eliminating time from $$x(t)$$ and $$y(t)$$ gives a parabola. The form with $\tan\alpha$ is a quadratic in $\tan\alpha$ , which is how the angle to hit a given point is found.

Appears in problems (2) 1.3.6 1.3.13

Relative motion of two bodies in gravity Относительное движение двух тел в поле тяжести

method 1.3

отнотнотн$$\vec v_{\text{отн}} = \text{const}, \qquad \vec r_{\text{отн}} = \vec r_{0} + \vec v_{\text{отн}}\, t$$

$отн$\vec v_{\text{отн}}$$: velocity of one body relative to the other
$отн$\vec r_{\text{отн}}$$: vector from one body to the other

Both bodies have the same acceleration $$g$$ , so their relative velocity does not change and one moves relative to the other uniformly in a straight line. The distance between them is found as in a problem without gravity.

Appears in problems (2) 1.3.1 1.3.14

Radius of curvature of a trajectory Радиус кривизны траектории

method 1.3

R = \frac{v^2}{a_n}

ввершинепараболы$\displaystyle R = \frac{v_0^2\cos^2\alpha}{g}\ \text{(в вершине параболы)}$

вточкегдескоростьнаклоненаподуглом$\displaystyle R = \frac{v^2}{g\cos\varphi}\ \text{(в точке, где скорость наклонена под углом }\varphi)$

R: radius of the circle touching the path at that point
a_n: normal acceleration there
v: speed there

The radius of curvature follows from the speed and the normal component of the acceleration at the point. For a projectile the acceleration is $$g$$ and its normal part is $g\cos\varphi$ with $\varphi$ the slope of the velocity, so at the top of the parabola $$R = v_x^2/g$$ .

Appears in problems (2) 1.3.28 1.5.8

Envelope of trajectories Парабола безопасности

law 1.3

y = \frac{v_0^2}{2g} - \frac{g x^2}{2 v_0^2}

\displaystyle v_{\min}^2 = g\left(y + \sqrt{x^2 + y^2}\right)

v₀: initial speed
$v_{\min}$: the least speed that reaches the point $$(x, y)$$

Every trajectory with the same initial speed lies inside this parabola and points outside it cannot be reached. The tangency condition is a vanishing discriminant of the trajectory equation in $\tan\alpha$ . The same gives the least speed to hit a given point.

Appears in problems (1) 1.3.13

1.4Galileo's transformations

Addition of velocities Сложение скоростей

law 1.4

\vec v = \vec v\,' + \vec u

отн$\displaystyle \vec v_{\text{отн}} = \vec v_1 - \vec v_2$

отн$\displaystyle v_{\text{отн}} = \sqrt{v_1^2 + v_2^2 - 2 v_1 v_2\cos\alpha}$

$\vec v$: velocity relative to the fixed frame
$\vec v\,'$: velocity relative to the moving frame
$\vec u$: velocity of the moving frame

A body's velocity relative to the ground is the vector sum of its velocity relative to a moving frame and the frame's own velocity. The relative velocity of two bodies is the difference of their velocities. A problem is often easiest in the frame where one body is at rest.

Triangle of velocities Треугольник скоростей

method 1.4

\sin\alpha = \frac{u}{v}

отн$\displaystyle v_{\text{отн}} = \sqrt{v^2 - u^2}$

\displaystyle \tan\varphi = \frac{u}{v}

v: speed relative to the medium, a boat relative to the water
u: speed of the medium, the current or the wind
$\alpha$: angle between the velocity and the required direction

The velocity relative to the ground, the velocity relative to the medium and the medium's velocity form a triangle. To cross a river straight, the bow is turned upstream so that $v\sin\alpha = u$ . The shortest crossing is with the velocity perpendicular to the bank.

Reflection from a moving wall Отражение от движущейся стенки

law 1.4

$$v' = -v + 2u$$

\displaystyle \Delta v = -2(v - u)

стенкадвижетсянавстречу$\displaystyle v' = v + 2u\ \text{(стенка движется навстречу)}$

v: speed before the hit, normal to the wall
u: speed of the wall
$$v'$$: velocity after the hit

In the wall's frame the collision is elastic and the normal velocity component simply flips sign. Back in the fixed frame the body leaves an approaching wall faster by $$2u$$ and a receding wall slower by $$2u$$ . The tangential component is unchanged.

Appears in problems (3) 1.4.8 1.4.9 1.4.12

Galilean transformation Преобразование Галилея

law 1.4

\vec r\,' = \vec r - \vec u\,t, \qquad t' = t

\displaystyle \vec v\,' = \vec v - \vec u

\displaystyle \vec a\,' = \vec a

$\vec r, \vec r\,'$: position vectors in the fixed and the moving frame
$\vec u$: constant velocity of the moving frame

Passing to a frame moving uniformly in a straight line subtracts $\vec u t$ from positions and $\vec u$ from velocities, and leaves accelerations and time intervals unchanged. The laws of mechanics are the same in every such frame.

Appears in problems (1) 1.4.3

1.5Motion with constraints

Constraint of an inextensible rope Связь через нерастяжимую нить

method 1.5

v_1\cos\alpha_1 = v_2\cos\alpha_2

\displaystyle u = \frac{v}{\cos\alpha}

v₁, v₂: velocities of the rope's ends
$\alpha_1, \alpha_2$: angles between the velocities and the rope

An inextensible rope can neither stretch nor shorten, so the projections of its ends' velocities on the rope are equal. This links the speeds of loads, pulleys and points of a rod, where the same holds for the ends of the rod.

Appears in problems (19) 1.1.12 1.3.6 1.3.27 1.3.28 1.3.30 1.5.7 1.5.12 1.5.13

Rolling without slipping Качение без проскальзывания

law 1.5

верхниз$$v_O = \omega R, \qquad v_{\text{верх}} = 2v_O, \qquad v_{\text{низ}} = 0$$

\displaystyle \vec v_A = \vec v_O + \vec\omega\times\vec r_A

\displaystyle v_A = 2v_O\cos\frac{\varphi}{2}

v_O: speed of the rolling body's centre
$\omega$: angular velocity
R: radius
v_A: speed of a rim point, $\varphi$ its angle from the bottom

The contact point with the support is at rest, so the centre moves at $\omega R$ and the top point twice as fast. Any point's velocity is the centre's velocity plus the velocity of rotation about the centre. On a moving support the support's velocity is added.

Appears in problems (10) 1.5.2 1.5.8 1.5.12 2.3.24 2.6.37 2.7.34 2.7.47 2.8.37

Instantaneous centre of rotation Мгновенный центр скоростей

method 1.5

v_A = \omega\,|CA|

C: the point whose velocity is zero at that moment
$$|CA|$$: its distance to the point $$A$$
$\omega$: angular velocity of the body

At every moment the plane motion of a rigid body is a rotation about the point whose velocity is zero. It lies where the perpendiculars to the velocities of two points meet, and any point moves at $\omega$ times its distance to that centre. That distance is not the radius of curvature of the point's path, because the centre itself moves.

Appears in problems (4) 1.5.9 1.5.12 1.5.17 1.5.18

1. KinematicsSavchenko Formulas, chapter 1 of 14, 31 formulas

1.1Constant-speed motion

Uniform motion Равномерное движение

Average speed Средняя скорость

Closing and separating speed Скорость сближения и удаления

Velocity from its components Скорость по компонентам

1.2Variable-speed motion

Uniformly accelerated motion Равноускоренное движение

Velocity as the derivative of the coordinate Скорость как производная координаты

Acceleration as the derivative of velocity Ускорение как производная скорости

Distance as the integral of velocity Путь как интеграл скорости

Volume flow rate Объёмный расход

Meeting condition Условие встречи

1.3Motion in gravity field. Curvilinear motion

Centripetal acceleration Центростремительное ускорение

Angular velocity and period Угловая скорость и период

Free fall Свободное падение

Projectile motion Тело, брошенное под углом к горизонту

Tangential and normal acceleration Тангенциальное и нормальное ускорения

Velocity components of a projectile Скорость брошенного тела по компонентам

Maximum height and time of flight Высота и время полёта

Acceleration along a smooth incline Ускорение вдоль гладкой наклонной плоскости

Range of a projectile Дальность полёта

Trajectory equation Уравнение траектории

Relative motion of two bodies in gravity Относительное движение двух тел в поле тяжести

Radius of curvature of a trajectory Радиус кривизны траектории

Envelope of trajectories Парабола безопасности

1.4Galileo's transformations

Addition of velocities Сложение скоростей

Triangle of velocities Треугольник скоростей

Reflection from a moving wall Отражение от движущейся стенки

Galilean transformation Преобразование Галилея

1.5Motion with constraints

Constraint of an inextensible rope Связь через нерастяжимую нить

Rolling without slipping Качение без проскальзывания

Instantaneous centre of rotation Мгновенный центр скоростей

Differentiating a constraint Дифференцирование связи