CALC A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is F ext = d p d t = m d υ d t + υ d m d t Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m = kx , where k is a constant, and dm/dt = kυ . This gives, since F ext = mg , m g = m d υ d t + υ ( k υ ) Or, dividing by k , x g = x d υ d t + υ 2 This is a differential equation that has a solution of the form υ = at , where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero, (a) Using the proposed solution for v , find the acceleration a . (b) Find the distance the rain-drop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics , Vol. 49 (1981), pp. 113–117.)
CALC A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is F ext = d p d t = m d υ d t + υ d m d t Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m = kx , where k is a constant, and dm/dt = kυ . This gives, since F ext = mg , m g = m d υ d t + υ ( k υ ) Or, dividing by k , x g = x d υ d t + υ 2 This is a differential equation that has a solution of the form υ = at , where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero, (a) Using the proposed solution for v , find the acceleration a . (b) Find the distance the rain-drop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics , Vol. 49 (1981), pp. 113–117.)
CALC A Variable-Mass Raindrop. In a rocket-propulsion problem the mass is variable. Another such problem is a raindrop falling through a cloud of small water droplets. Some of these small droplets adhere to the raindrop, thereby increasing its mass as it falls. The force on the raindrop is
F
ext
=
d
p
d
t
=
m
d
υ
d
t
+
υ
d
m
d
t
Suppose the mass of the raindrop depends on the distance x that it has fallen. Then m = kx, where k is a constant, and dm/dt = kυ. This gives, since Fext = mg,
m
g
=
m
d
υ
d
t
+
υ
(
k
υ
)
Or, dividing by k,
x
g
=
x
d
υ
d
t
+
υ
2
This is a differential equation that has a solution of the form υ = at, where a is the acceleration and is constant. Take the initial velocity of the raindrop to be zero, (a) Using the proposed solution for v, find the acceleration a. (b) Find the distance the rain-drop has fallen in t = 3.00 s. (c) Given that k = 2.00 g/m, find the mass of the raindrop at t = 3.00 s. (For many more intriguing aspects of this problem, see K. S. Krane, American Journal of Physics, Vol. 49 (1981), pp. 113–117.)
Answer must be in scientific notation with SI units that do not have prefixes except for kg. (m/s not cm/s). Answer must be in standard form scientific notation. All angles are to be calculated to the nearest 0.1 deg (tenth of a degree).
A cat of 20 kg mass is running at 8.0 m/s toward a stationary skateboard with a mass of 2 kg. The cat jumps on the skateboard once it reaches it. The skateboard and cat roll forward together without friction until they reach a ramp 30◦ above horizontal. If the coefficient of kinetic friction between the skateboard and the ramp is µk =0.3, what is the maximum height the skateboard can reach?
The coefficient of friction between the block of mass m1 = 3.00 kg and the surface is µk = 0.400.
The system starts from rest. What is the speed of the ball of mass m2 = 5.00 kg when it has fallen a
2.
distance h=1.50 m?
m1
M2
Chapter 8 Solutions
University Physics with Modern Physics, Volume 1 (Chs. 1-20) (14th Edition)
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