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
(1.1b)
v— dt ' w = dt
• Acceleration of a point expressed in projections on the tangent and the
normal to a trajectory:
dv,
V2
wt =
—
(1.1c)
dt ' w n
R '
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:
dtp
do)
cu
dt
clt
(1.1e)
• Relation between linear and angular quantities for a rotating solid
body:
v = [ow], wn = o)2R, 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 v0. The
remaining part of the distance was covered with velocity vl for half
the time, and with velocity v2 for the other half of the time. Find
the mean velocity of the point averaged over the whole time of motion.