– Average vectors of velocity and acceleration of a point:
\[
\langle\mathbf{v}\rangle=\frac{\Delta \mathbf{r}}{\Delta t}, \quad\langle\mathbf{w}\rangle=\frac{\Delta \mathbf{v}}{\Delta t},
\]
where $\Delta \mathbf{r}$ is the displacement vector (an increment of a radius vector).
– Velocity and acceleration of a point:
\[
\mathbf{v}=\frac{d \mathbf{r}}{d t}, \quad \mathbf{w}=\frac{d \mathbf{v}}{d t} .
\]
– Acceleration of a point expressed in projections on the tangent and the normal to a trajectory:
\[
w_{\tau}=\frac{d v_{\tau}}{d t}, \quad w_{n}=\frac{v^{2}}{R},
\]
where $R$ is the radius of curvature of the trajectory at the given point.
– Distance covered by a point:
\[
s=\int v d t
\]
where $v$ is the modulus of the velocity vector of a point.
– Angular velocity and angular acceleration of a solid body:
\[
\omega=\frac{d \varphi}{d t}, \quad \beta=\frac{d \omega}{d t} .
\]
– Relation between linear and angular quantities for a rotating solid body:
\[
\mathbf{v}=[\boldsymbol{\omega}], \quad w_{n}=\omega^{2} R, \quad\left|w_{\tau}\right|=\beta R,
\]
where $\mathbf{r}$ is the radius vector of the considered point relative to an arbitrary point on the rotation axis, and $R$ is the distance from the rotation axis.
1.1. A motorboat going downstream overcame a raft at a point $A$; $\tau=60 \mathrm{~min}$ later it turned back and after some time passed the raft at a distance $l=6.0 \mathrm{~km}$ from the point $A$. Find the flow velocity assuming the duty of the engine to be constant.
1.2. A point traversed half the distance with a velocity $v_{0}$. The remaining part of the distance was covered with velocity $v_{1}$ for half the time, and with velocity $v_{2}$ for the other half of the time. Find the mean velocity of the point averaged over the whole time of motion.
1.3. A car starts moving rectilinearly, first with acceleration $w=$ $=5.0 \mathrm{~m} / \mathrm{s}^{2}$ (the initial velocity is equal to zero), then uniformly, and finally, decelerating at the same rate $w$, comes to a stop. The total time of motion equals $\mathrm{z}=25 \mathrm{~s}$. The sverage velocity during that time is equal to $(v)=72 \mathrm{~km}$ per hour. How long does the car move uniformly?
1.4. A point moves rectilinearly is ene direction. Fig. 1.1 shows
Fie. 1.1.
the distance s traversed by the point as a function of the time $t$. Using the plot find:
(a) the average velocity of the point daring the time of motion:
(b) the maximum velocity;
(c) the time moment $t_{6}$ at which the instantaneous velocity is equal to the mean velocity averaged over the first $t_{\text {, seconds. }}$
1.5. Two particles, $l$ and 2 , move with constant velocities $\mathrm{v}_{1}$ and v. At the initial moment their radius vectors are equal to $r_{1}$ and $r_{7}$ ? How must these four vectors be interrelated for the particles to collide?
1.6. A ship moves along the equator to the east with velocity $v_{0}=30 \mathrm{~km} /$ hour. The southeasters wind blows at an angle $\varphi=60^{\circ}$ to the equator with velocity $v=15 \mathrm{~km} /$ hour. Find the wind velocity of relative to the ship and the angle $\varphi^{\prime}$ between the equator and the wind direction in the reference frame fixed to the ship.
1.7. Two swimmers leave poist $A$ os one bank of the river to reach point $B$ lying right across on the other bank. One of them crosses the river along the straight line $A B$ while the other swims at right angles to the stream and then walks the distance that he has been carried away by the stream to get to point $B$. What was the velocity $u$
of his walking if beth swimmers reached the destination simulta: neously? The stream velocity $v_{0}=2.0 \mathrm{~km} /$ hour and the velocity $v^{\prime}$ of each swimmer with respect to water equals $2.5 \mathrm{~km}$ per hour.
1.8. Two boats, $\boldsymbol{A}$ and $\boldsymbol{B}$, move away from a buoy anchered at the middle of a river along the mutually perpendicular straight lines: the boat $A$ along the river, and the boat $B$ across the river. Having moved off an equal distance from the buey the boats returned. Find the ratio of times of motion of boats $\tau_{A} / \tau_{n}$, if the velocity of aach beat with respect to water is $\eta=1.2$ times greater than the stream velocity.
1.9. A boat moves relative to water with a velocity which is $n=$ $=2.0$ times less than the river flow velocity. At what angle to the stream direction must the boat move to minimise drifting?
1.10. Two bodies were thrown simultaneously from the same point: one, straight up, and the other, at an angle of $\theta=60^{\circ}$ to the horizontal. The initial velocity of each body is equal to $v_{0}=25 \mathrm{~m} / \mathrm{s}$. Neglecting the air drag, find the distance between the bodies $t=$ $=1.70$ s later.
1.11. Twe particles move in a uniform gravitational feld with an acceleration $\mathrm{g}$. At the initial moment the particles were located at one point and moved with velocities $y_{1}=3.0 \mathrm{~m} / \mathrm{s}$ and $v_{3}=4.0 \mathrm{~m} / \mathrm{s}$ horizontally in opposite directions. Find the distance between the particles at the moment when their velocity vectors become mutually perpendicular.
1.12. Thre points are located at the vertices of an equilateral triangle whose side equals $a$. They all start moving simultaneously with velocity o constant in medulus, with the first point heading continually for the second, the second for the third, and the third for the first. How soon will the points converge?
1.13. Point $\boldsymbol{A}$ moves uniformly with velocity $v$ so that the vector $\mathrm{v}$ is continually \”aimed\” at point $B$ which in its turn moves rectilinearly and uniformly with velocity $u<v$. At the initial moment of time $\mathrm{v} \perp \mathrm{u}$ and the points are separated by a distance $L$. How soos will the points converge?
1.14. A train of length $l-350 \mathrm{~m}$ starts moving rectilinearly with constant acceleration $w=3.0 \cdot 10^{-4} \mathrm{~m} / \mathrm{s}^{2} ; t=30$ s after the start the locomotive headlight is switched on (event $D$ ), and $\tau=60$, after that event the tail signal light is switched on (event 2 ). Find the distance between these events in the reference trames fixed to the trais and to the Earth. How and at what constant velocity $\boldsymbol{V}$ relstive to the Earth must a certain reference frame $K$ move for the two events to occur is it at the same point?
1.15. An elevator car whose floor-to-ceiling distance is equal to $2.7 \mathrm{~m}$ starts ascending with constant acceleration $1.2 \mathrm{~m} / \mathrm{s}^{2} ; 2.0$, after the start a bolt begins falling from the ceiling of the car. Find:
(a) the bolt’s free fall time;
(b) the displacement and the distance covered by the bolt during the tree fall in the reference frame fixed to the elevater shaft.
1.16. Two particles, $\boldsymbol{I}$ and 2 , move with constant velocities on and $v$, along twe mutually perpendicular straight lines toward the intersection point $O$. At the moment $t=0$ the particles were located at the distances $h_{1}$ and $l_{2}$ from the point $O$. How soon will the distance between the particles become the smallest? What is it equal to?
1.17. From point $A$ located on a highway (Fig. 1.2) one has to get by car as soon as possible to point $B$ located in the field at a distance $l$ from the highway. It is knows that the ear moves in the feld $\eta$ times slower than on the highway. At what distance from point $\bar{D}$ one must turs of the highway?
1.18. A point travels along the $x$ axis with a velocity whose projection $v_{x}$ is presented as a function of time by the plot in Fig. 1.3.
Fie. 1.2.
Fie. 1.3.

Asuming the coordinate of the point $x=0$ at the mement $t=0$, draw the approximate time dependence plots for the acceleration $w_{n}$. the $x$ coordinate, and the distance covereds.
1.19. A point traversed halt a circle of radius $\boldsymbol{R}=\mathbf{1 6 0} \mathrm{cm}$ during time interval $\tau=10.0$ s. Calculate the following quantities averaged over that time:
(a) the mean velocity (v);
(b) the modulus of the mean velocity vecter $|(\mathbf{v})|$;
(c) the modulus of the mean vector of the total acceleration $|(w)|$ if the point moved with constant tangent acceleration.
1.20. A radius vector of a particle varies with time $t$ as $r=$ = at $(1-a t)$, where $a$ is a constant vector and $\alpha$ is a positive factor. Find:
(a) the velocity $\mathrm{v}$ and the acceleration $\mathrm{w}$ of the particle as functions of time:
(b) the time interval $\Delta t$ taken by the particle to return to the initial points, and the distance $s$ covered during that time.
1.21. At the momeat $t=0$ a particle leaves the origin and moves in the positive direction of the $z$ axis. Its velocity varies with time as $=\mathrm{v}_{0}(1-t / \mathrm{v})$, where $\mathrm{v}_{0}$ is the initial velocity vector whose modulus equals $v_{0}=10.0 \mathrm{~cm} / \mathrm{s} ; \tau=5.0 \mathrm{~s}$. Find:
(a) the $x$ coordinate of the particle at the moments of time 6.0 , 10 , and $20 \mathrm{~s} ;$
(b) the moments of time when the particle is at the distance $10.0 \mathrm{~cm}$ from the origin;
(c) the distance $s$ covered by the particle during the first 4.0 and $8.0 \mathrm{~s}$ draw the approximate plot $s(f)$.
1.2. The velocity of a partiele moving in the positive direction of the $x$ axis varies as $y=a \sqrt{x}$, where $a$ is a positive constant. Assuming that at the moment $t=0$ the particle was located at the point $x=0$, find:
(a) the time dependence of the velocity and the acceleration of the particle:
(b) the mean velocity of the particle averaged over the time that the particle takes to cever the first $s$ metres of the path.
1.23. A point moves rectilinearly with deceleration whose modulus depends on the velocity $v$ of the particle as $w=a V \bar{v}$, where $a$ is a positive constant. At the initial moment the velocity of the point is equal to $v_{0}$. What distance will it traverse before it stops? What time will it take to cever that distance?
1.24. A radius vector of a point $A$ relative to the origin varies with time $t$ as $r=a f l-b t^{2} j$, where $a$ and $b$ are positive constants, and $i$ and $j$ are the unit vecters of the $x$ and $y$ axes. Find:
(a) the equation of the point’s trajectery $y(x)$; plot this function;
(b) the time dependence of the velocity $\mathbf{v}$ and acceleration $\mathbf{v e c}$ tors, as well as of the moduli of these quantities:
(c) the time dependence of the angle $a$ between the vectors $w$ and $v$;
(d) the mean velocity vector averaged over the first $t$ seconds of motion, and the modulus of this vector.
1.25. A point moves in the plane $x y$ according to the law $x=$ at, $y=a t(1-a f)$, where $a$ and $\alpha$ are positive constants, and $t$ is time. Find:
(a) the equation of the point’s trajectory $y(x)$; plot this function;
(b) the velocity $v$ and the acceleration $w$ of the point as functions of time:
(c) the moment $t_{0}$ at which the velocity vector forms an angle $a / 4$ with the acceleration vector.
1.26. A point moves in the plane $x y$ according to the law $x=$ $=a$ sin $e t, y=a(1-\cos e f)$, where $a$ and $o$ are positive constants. Find:
(a) the distance $s$ traversed by the point during the time $\mathrm{x}$;
(b) the angle between the point’s velocity and acceleration vectors.
1.27. A particle moves in the plane $x y$ with constant acceleration w directed along the negative $y$ axis. The equation of motion of the particle has the form $y-a x-b x^{2}$, where $a$ and $b$ are positive constants. Find the velocity of the particle at the origin of coordinates.
1.28. A small body is thrown at an angle to the herizontal with the initial velocity $\mathbf{v}_{0}$. Neglecting the air drag, find:
(a) the displacement of the bedy as a function of time $r(t)$;
(b) the mean velocity vector (v) averaged over the first $t$ seconds and over the total time of motion.
1.29. A body is thrown from the surface of the Earth at an angle $a$ to the horizontal with the initial velocity $v_{8}$. Assuming the air drag to be negligible, find:
(a) the time of motion:
(b) the maximum height of ascent and the horizontal range; at what value of the angle $a$ they will be equal to each other;
(c) the equation of trajectory $y(x)$, where $y$ and $z$ are displacements of the body along the vertical and the herizontal respectively;
(d) the curvature radii of trajectery at its initial point and at its peak.
1.30. Using the conditions of the foregoing problem, draw the approximate time dependence of moduli of the normal $w$, and tangent $w_{4}$ acceleration vectors, as well as of the projection of the total acceleration vector $w$, on the velocity vector direction.
1.31. A ball starts falling with sero initial velocity on a smooth inclined plane forming an angle $a$ with the horisontal. Having fallen the distance $h$, the ball rebounds elastically of the inclined plane. At what distance from the impact point will the ball rebound for the second time?
1.32. A cannon and a target are $5.10 \mathrm{~km}$ apart and located at the same level. How soon will the shell launched with the initial velocity $240 \mathrm{~m} / \mathrm{s}$ reach the target in the absence of air drag?
1.33. A cannon fires successively two shells with velecity $v_{0}=$ $-250 \mathrm{~m} / \mathrm{s}$, the first at the angle $6_{1}=60^{\circ}$ and the second at the angle $\theta_{2}=45^{\circ}$ to the horisontal, the mimuth being the same. Neglecting the air drag, find the time interval between firings leading to the collision of the shells.
1.34. A balloon starts rising from the surface of the Earth. The ascension rate is constant and equal to $F_{6}$. Due to the wind the balloon gathers the herizontal velocity component $v_{z}=a y$, where a is a constant and $y$ is the height of ascent. Find how the following quantities depend on the height of ascent:
(a) the horizontal drift of the balloon $x(y)$;
(b) the total, tangential, and normal accelerations of the balloon.
1.35. A particle moves in the plane $x y$ with velocity $\mathrm{v}=\mathrm{al}+b x$. where $i$ and $\}$ are the unit vectors of the $z$ and $y$ axes, and $a$ and $b$ are constants. At the initial moment of time the particle was located at the point $x=y=0$. Find:
(a) the equation of the particle’s trajectory $y(x)$;
(b) the curvature radius of trajectory as a function of $x$.
1.36. A particle $A$ moves in one direction along a given trajectory with a taagential acceleration $w_{y}=\mathbf{a r}$, where a is a constant vector coinciding in direction with the $x$ axis (Fig. 1.4), and $\tau$ is a unit vector coinciding in direction with the velocity vector at a given point. Find how the velocity of the particle depends on $x$ provided that its velocity is negligible at the point $x=0$.
1.37. A point moves along a circle with a velocity $v=a t$, where $a=0.50 \mathrm{~m} / \mathrm{s}^{2}$. Find the total acceleration of the point at the me16
ment when it covered the $n$-th $(n=0.10)$ fraction of the cirele after the beginning of motion.
1.38. A point moves with deceleration along the circle of radius $R$ so that at any moment of time its tangential and normal accelerations
Fie. 1.4.
are equal in moduli. At the initial mement $t=0$ the velocity of the point equals $v_{4}$. Find:
(a) the velocity of the point as a function of time and as a function of the distance cevered $s_{5}$
(b) the total acceleration of the point as a function of velocity and the distance cevered.
1.39. A point moves along an arc of a circle of radius $A$. Its velecity depends on the distance cevered $s$ as $v=a \sqrt{3}$, where $a$ is a constant. Find the angle $a$ between the vecter of the total acceleration and the vecter of velocity as a function of $s$.
1.40. A particle moves along an are of a circle of radius $A$ aconding to the law $l=a$ sis $\omega t$, where $l$ is the displocement from the initial pesition measured along the are, and a and e are constants. Assuming $R=1.00 \mathrm{~m}, a=0.80 \mathrm{~m}$, and $\omega=2.00 \mathrm{rad} / \mathrm{s}$, find:
(a) the magnitude of the tetal acceleration of the particle at the points $l=0$ and $l= \pm a$;
(b) the minimum value of the total acceleration $w_{\text {mis }}$ and the corresponding displacement $i_{m}$.
1.41. A point moves in the plane so that its tangential acceleration $w_{4}=a$, and its normal acoleration $w_{n}=b r^{4}$, where $a$ and $b$ are positive constants, and $t$ is time. At the moment $t=0$ the point was at rest. Find how the curvature radius $\boldsymbol{R}$ of the point’s trajectory and the total acceleration $w$ depend on the distance covered $s$.
1.42. A particle moves along the plane trajectery $y(x)$ with velocity $v$ whose modulus is constant. Find the acceleration of the particle at the point $x=0$ and the curvature radius of the trajectory at that point if the trajectory has the form
(a) of a parabele $y=\Delta x^{2}$;
(b) of an ellipse $(x / a)^{2}+\left(y^{\prime} b\right)^{2}-1 ; a$ and $b$ are constants here.
1.43. A particle $\boldsymbol{A}$ moves along a circle of radius $\boldsymbol{R}=50 \mathrm{~cm}$ so that its radius vecter $r$ relative to the point $O$ (Fig. 1.5) rotates with the constant angular velecity $0=0.40 \mathrm{rad} / \mathrm{s}$. Find the modulus of the velocity of the particle, and the modulus and direction of its total scceleration.
1.44. A wheel rotates around a stationary axis so that the rotation angle $\varphi$ varies with time as $\varphi=a t^{2}$, where $a=0.20 \mathrm{rad} / \mathrm{s}^{2}$. Find the total acceleration $w$ of the point $A$ at the rim at the moment $t=2.5$, if the linear velocity of the point $A$ at this moment $v=0.65 \mathrm{~m} / \mathrm{s}$.
1.45. A shell acquires the initial velocity $v=320 \mathrm{~m} / \mathrm{s}$, having made $n-2.0$ turns inside the barrel whese length is equal to $l=$ $=2.0 \mathrm{~m}$. Assuming that the shell moves inside the barrel with a usiform accelers. tion, find the angular velocity of its axial rotation at the moment when the shell escapes the barrel.
1.46. A solid body rotates about a stationary axis according to the law $\varphi=$ at $-b r^{2}$, where $a=6.0 \mathrm{rad} / \mathrm{s}$ and $b=2.0$ $\mathrm{rad} / \mathrm{s}^{2}$. Find:
(a) the mean values of the angular veloeity and angular acceleration averaged over the time interval between $t=0$ and the complete stop:
(b) the angular acceleration at the moment when the body stops.
1.47. A solid body starts rotating about a stationary axis with an angular acceleration $\beta=$ at, where $a=2.0 \cdot 10^{-4} \mathrm{rad} / \mathrm{s}^{3}$. How soon after the beginning of rotation will the total acceleration vector of an arbitrary point of the body form an angle $\alpha=60^{\circ}$ with its velocity vector?
1.48. A solid body rotates with deceleration about a stationary axis with an angular deceleration $\beta \propto \sqrt{\omega}$, where $\omega$ is its angular velocity. Find the mean angular velocity of the body averaged over the whole time of rotation if at the initial moment of time its angular velocity was equal to $\oplus_{0}$.
1.49. A solid body rotates about a stationary axis so that its angular velocity depends on the rotation angle $\&$ as $6=\omega_{4}-a \varphi$, where $\omega$, and a are positive constants. At the moment $t=0$ the angle $\varphi=0$. Find the time dependence of
(a) the rotation angle;
(b) the angular velocity.
15e. A solid body starts rotating about a stationary axis with an angular acceleration $\beta-\beta_{0} \cos \psi$, where $\beta_{0}$ is a constant vector and $\psi$ is an angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle 5 . Draw the plot of this dependence.
1.51. A rotating dise (Fig. 1.6) moves in the positive direction of the $x$ axis. Find the equation $y(x)$ describing the position of the instantaneous axis of rotation, if at the initial moment the axis $C$ of the dise was located at the point $O$ after which it moved
(a) with a constant velecity $v$, while the disc started rotating counterclockwise with a constant angular acceleration $\beta$ (the initial angular velocity is equal to zero);
18
(b) with a constant acceleration $w$ (and the zero initial velocity), while the dise rotates countercleckwise with a constant angular veloeity 6.
1.52. A point $A$ is lecated on the rim of a wheel of radius $R=$ $=0.50 \mathrm{~m}$ which rolls without slipping along a herizental surface with velecity $v=1.00 \mathrm{~m} / \mathrm{s}$. Find:
(a) the medulus and the direction of the acoeleration vector of the point A;
(b) the total distance $s$ traversed by the point $\boldsymbol{A}$ between the two successive moments at which it touches the surface.
1.53. A ball of radius $R=10.0 \mathrm{~cm}$ rolls without slipping down an inclined plane so that its centre meves with censtant acceleration
Fie. 1.6.
Fe. 1.7.
$w=2.50 \mathrm{~cm} / \mathrm{s}^{2} ; t=2.00 \mathrm{~s}$ after the beginning of motion its position corresponds to that shows is Fig. 1.7. Find:
(a) the velocities of the peints $A, B$, and $O$;
(b) the accelerations of these points.
1.54. A cylinder rolls without slipping over a horizontal plane. The radius of the cylinder is equal to $r$. Find the curvature radii of trajectories traced out by the points $A$ and $B$ (see Fig. 1.7).
1.55. Two solid bodies rotate about stationary mutualiy perpesdicular intersecting axes with constant angular velocities $\omega_{1}=$ $=3.0 \mathrm{rad} / \mathrm{s}$ and $\omega_{2}=4.0 \mathrm{rad} / \mathrm{s}$. Find the angular velocity and angular acceleration of one body relative to the other.
1.56. A solid body rotates with angular velecity $\omega=a t i+b t^{2}$. where $a=0.50 \mathrm{rad} / \mathrm{s}^{2}, b=0.060 \mathrm{rad} / \mathrm{s}^{2}$, and $i$ and $\}$ are the unit vectors of the $x$ and $y$ axes. Find:
(a) the moduli of the angular velocity and the angular acceleration at the moment $t=10.0 \mathrm{~s}$
(b) the angle between the vecters of the angular velocity and the angular scceleration at that moment.
1.57. A round cose with half-angle $a=30^{\circ}$ and the radius of the base $R=5.0 \mathrm{~cm}$ rolls uniformly and without slipping over a horizontal plane as shewn in Fig. 1.8. The cone apex is hinged at the peint $O$ which is on the same level with the point $C$, the cone base centre. The velocity of point $C$ is $v=10.0 \mathrm{~cm} / \mathrm{s}$. Find the moduli of

(a) the vector of the angular velocity of the one and the angle it forms with the vertical;
(b) the vector of the angular acceleration of the cone.
1.58. A solid body rotates with a constant angular velocity $\omega_{4}=$ $=0.50 \mathrm{rad} / \mathrm{s}$ about a horizental sxis $\boldsymbol{A B}$. At the moment $t=0$
Fig. 1.8.
the axis $A B$ starts turning about the vertical with a constant angular acceleration $\beta_{0}=0.10 \mathrm{rad} / \mathrm{s}^{2}$. Find the angular velocity and angular acceleration of the body after $t=3.5 \mathrm{~s}$.