to the horizontal with the initial velocity v0. 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 x are displacements
of the body along the vertical and the horizontal respectively;
(d) the curvature radii of trajectory 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 Lyn and tangent iv,
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 zero initial velocity on a smooth
inclined plane forming an angle a with the horizontal. Having fallen the distance h, the ball rebounds elastically off 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 km apart and located at the
same level. How soon will the shell launched with the initial velocity
240 m/s reach the target in the absence of air drag?
1.33. A cannon fires successively two shells with velocity vo =
= 250 m/s; the first at the angle 01 = 60° and the second at the angle
0 2 = 45° to the horizontal, the azimuth 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 vo. Due to the wind the balloon gathers the horizontal velocity component vx = ay, 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 xy with velocity v = ai
bxj,
where i and j are the unit vectors of the x 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 tangential acceleration u), = at, where a is a constant vector
coinciding in direction with the x axis (Fig. 1.4), and T 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 = at, where
a = 0.50 m/s2. Find the total acceleration of the point at the mo16
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 x are displacements
of the body along the vertical and the horizontal respectively;
(d) the curvature radii of trajectory 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 Lyn and tangent iv,
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 zero initial velocity on a smooth
inclined plane forming an angle a with the horizontal. Having fallen the distance h, the ball rebounds elastically off 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 km apart and located at the
same level. How soon will the shell launched with the initial velocity
240 m/s reach the target in the absence of air drag?
1.33. A cannon fires successively two shells with velocity vo =
= 250 m/s; the first at the angle 01 = 60° and the second at the angle
0 2 = 45° to the horizontal, the azimuth 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 vo. Due to the wind the balloon gathers the horizontal velocity component vx = ay, 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 xy with velocity v = ai
bxj,
where i and j are the unit vectors of the x 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 tangential acceleration u), = at, where a is a constant vector
coinciding in direction with the x axis (Fig. 1.4), and T 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 = at, where
a = 0.50 m/s2. Find the total acceleration of the point at the mo16