Solutions (^167)
In the stationary reference frame, the veloci-
ties of the a-particle and the proton are represent-
ed in the figure by vectors 0C 2 = OB' v' and
0C 1 = OA' + v'.
In order to solve the problem, we must deter-
mine the maximum possible length of vector OC 1 ,
i.e. in the isosceles triangle OA'C 1 , we must deter-
mine the maximum possible length of the base
for constant values of the lateral sides. Obviously,
the maximum magnitude of 0C 1 is equal to
2 X (4/5) v 0 == 1.6v^0. This situation corresponds
to a central collision.
1.65. The tyres of a motorcar leave a trace in the
sand. The higher the pressure on the sand, the
deeper the trace, and the higher the probability
that the car gets stuck. If the tyres are deflated
considerably, the area of contact between the tyres
and the sand increases. In this case, the pressure
on the sand decreases, and the track becomes more
shallow.
1.66. At any instant of time, the complex motion
of the body in the pipe can be represented as the
superposition of two independent motions: the
motion along the axis of the pipe and the motion
in the circle in a plane perpendicular to the pipe
axis (Fig. 169). The separation of the body from
the pipe surface will affect only the latter motion
(the body will not move in a circle). Therefore,
we shall consider only this motion.
The body moving in a circle experiences the
action of the normal reaction N of the pipe walls
(vector N lies in the plane perpendicular to the
pipe axis) and the "force of gravity" mg' =--
mg cos a. We shall write the condition of motion
for the body in a circle:
mv 2
mg' cosii+ N = r ,
where p is the angle formed by the radius vector
of the point of location of the body at a given in-
stant and the "vertical" y' (Fig. 170). For the body
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