64
2 Particle Dynamics
Find
(a) the rocket thrust,
(b) net acceleration at the beginning,
(c) time to reach the burn-out velocity,
(d) the burn-out velocity.
2.63 A 5000 kg rocket is to be fired vertically. Calculate the rate of ejection of gas
at exhaust speed 100 m/s in order to provide necessary thrust to
(a) support the weight of the rocket and
(b) impart an initial upward acceleration of 2 g.
2.64 A flexible rope of length L and mass per unit length μ slides over the edge
of a frictionless table. Initially let a length y 0 of it be hanging at rest over the
edge and at time t let a length y moving with a velocity dy/dt be over the
edge. Obtain the equation of motion and discuss its solution.
2.65 An open railway car of mass W is running on smooth horizontal rails under
rain falling vertically down which it catches and retains in the car. If v 0 is the
initial velocity of the car and k the mass of rain falling into the car per unit
time, show that the distance travelled in time t is (W v 0 /k) ln(1 + kt/W ).
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.66 A heavy uniform chain of length L and mass M hangs vertically above a
horizontal table, its lower end just touching the table. When it falls freely,
show that the pressure on the table at any instant during the fall is three times
the weight of the portion on the table.
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.67 A spherical rain drop of radius R cm falls freely from rest. As it falls it accumulates condensed vapour proportional to its surface. Find its velocity when
it has fallen for t s.
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.3 Solutions
2.3.1 Motion of Blocks on a Plane
2.1 (a) Acceleration =
Force
Total mass
a =
F
(m 1 + m 2 + m 3 )
(1)
2 Particle Dynamics
Find
(a) the rocket thrust,
(b) net acceleration at the beginning,
(c) time to reach the burn-out velocity,
(d) the burn-out velocity.
2.63 A 5000 kg rocket is to be fired vertically. Calculate the rate of ejection of gas
at exhaust speed 100 m/s in order to provide necessary thrust to
(a) support the weight of the rocket and
(b) impart an initial upward acceleration of 2 g.
2.64 A flexible rope of length L and mass per unit length μ slides over the edge
of a frictionless table. Initially let a length y 0 of it be hanging at rest over the
edge and at time t let a length y moving with a velocity dy/dt be over the
edge. Obtain the equation of motion and discuss its solution.
2.65 An open railway car of mass W is running on smooth horizontal rails under
rain falling vertically down which it catches and retains in the car. If v 0 is the
initial velocity of the car and k the mass of rain falling into the car per unit
time, show that the distance travelled in time t is (W v 0 /k) ln(1 + kt/W ).
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.66 A heavy uniform chain of length L and mass M hangs vertically above a
horizontal table, its lower end just touching the table. When it falls freely,
show that the pressure on the table at any instant during the fall is three times
the weight of the portion on the table.
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.67 A spherical rain drop of radius R cm falls freely from rest. As it falls it accumulates condensed vapour proportional to its surface. Find its velocity when
it has fallen for t s.
[with courtesy from R.W. Norris and W. Seymour, Mechanics via
Calculus, Longmans, Green and Co., 1923]
2.3 Solutions
2.3.1 Motion of Blocks on a Plane
2.1 (a) Acceleration =
Force
Total mass
a =
F
(m 1 + m 2 + m 3 )
(1)