6
1 Kinematics and Statics
(i) Determine the magnitude and direction of the velocity of ship B relative
to ship A.
(ii) What will be their distance of closest approach?
[University of Manchester 2008]
1.20 A balloon is ascending at the rate of 9.8 m/s at a height of 98 m above the
ground when a packet is dropped. How long does it take the packet to reach
the ground?
1.2.2 Motion in Resisting Medium
1.21 An object of mass m is thrown vertically up. In the presence of heavy air
resistance the time of ascent (t 1 ) is no longer equal to the time of descent (t 2 ).
Similarly the initial speed (u) with which the body is thrown is not equal to the
final speed (v) with which the object returns. Assuming that the air resistance
F is constant show that
t 2
t 1
=
g + F/m
g − F/m
;
v
u
=
g − F/m
g + F/m
1.22 Determine the motion of a body falling under gravity, the resistance of air
being assumed proportional to the velocity.
1.23 Determine the motion of a body falling under gravity, the resistance of air
being assumed proportional to the square of the velocity.
1.24 A body is projected upward with initial velocity u against air resistance which
is assumed to be proportional to the square of velocity. Determine the height
to which the body will rise.
1.25 Under the assumption of the air resistance being proportional to the square
of velocity, find the loss in kinetic energy when the body has been projected
upward with velocity u and return to the point of projection.
1.2.3 Motion in Two Dimensions
1.26 A particle moving in the xy-plane has velocity components dx/dt = 6 + 2t
and dy/dt = 4 + t
where x and y are measured in metres and t in seconds.
(i) Integrate the above equation to obtain x and y as functions of time, given
that the particle was initially at the origin.
(ii) Write the velocity v of the particle in terms of the unit vectors ˆ
i and ˆ
j.
1 Kinematics and Statics
(i) Determine the magnitude and direction of the velocity of ship B relative
to ship A.
(ii) What will be their distance of closest approach?
[University of Manchester 2008]
1.20 A balloon is ascending at the rate of 9.8 m/s at a height of 98 m above the
ground when a packet is dropped. How long does it take the packet to reach
the ground?
1.2.2 Motion in Resisting Medium
1.21 An object of mass m is thrown vertically up. In the presence of heavy air
resistance the time of ascent (t 1 ) is no longer equal to the time of descent (t 2 ).
Similarly the initial speed (u) with which the body is thrown is not equal to the
final speed (v) with which the object returns. Assuming that the air resistance
F is constant show that
t 2
t 1
=
g + F/m
g − F/m
;
v
u
=
g − F/m
g + F/m
1.22 Determine the motion of a body falling under gravity, the resistance of air
being assumed proportional to the velocity.
1.23 Determine the motion of a body falling under gravity, the resistance of air
being assumed proportional to the square of the velocity.
1.24 A body is projected upward with initial velocity u against air resistance which
is assumed to be proportional to the square of velocity. Determine the height
to which the body will rise.
1.25 Under the assumption of the air resistance being proportional to the square
of velocity, find the loss in kinetic energy when the body has been projected
upward with velocity u and return to the point of projection.
1.2.3 Motion in Two Dimensions
1.26 A particle moving in the xy-plane has velocity components dx/dt = 6 + 2t
and dy/dt = 4 + t
where x and y are measured in metres and t in seconds.
(i) Integrate the above equation to obtain x and y as functions of time, given
that the particle was initially at the origin.
(ii) Write the velocity v of the particle in terms of the unit vectors ˆ
i and ˆ
j.