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.