## PART ONE. PHYSICAL FUNDAMENTALS OF MECHANICS

#### 1.1. Kinematics

- Average vectors of velocity and acceleration of a point:

`(vi At`

`, r A v`

(w)= (^) ' (1.1a)

where Ar is the displacement vector (an increment of a radius vector).

- Velocity and acceleration of a point:

dr dv

v— dt ' w = dt (1.1b) - Acceleration of a point expressed in projections on the tangent and the

normal to a trajectory:

`wt= dv, V^2`

dt ' w n — R ' (1.1c)

`where R is the radius of curvature of the trajectory at the given point.`

- Distance covered by a point:

`s =^ v dt,^ (1.1d)`

`where v is the modulus of the velocity vector of a point.`

- Angular velocity and angular acceleration of a solid body:

`cu dtp^ do)`

dt clt (1.1e)

- Relation between linear and angular quantities for a rotating solid

body:

`v = [ow], wn = o) 2 R, I w, I (1.1.f)`

where 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;

T = 60 min later it turned back and after some time passed the raft

at a distance 1 = 6.0 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 vl 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 mo-

tion.