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.)
Suppose you are navigating a spacecraft far from other objects. The mass of the spacecraft is 3.0 x 104 kg (about 30 tons). The rocket engines are shut off, and you're coasting along with
velocity of km/s. As you pass the location km you briefly fire side thruster rockets, so that your spacecraft experiences a net force of N for 18 s
ejected gases have a mass that is small compared to the mass of the spacecraft. You then continue coasting with the rocket engines turned off. Where are you an hour later? (Think about
approximations or simplifying assumptions you made in your analysis. Also think about the choice of system: what are the surroundings that exert external forces on your system?)
m
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
b)
Two students are watching an action film in which a car drives down a ramp onto the back of a
moving lory. Both are moving at high speed, the car slightly faster than the lorry.
One student complains that this is impossible because the car would not be able to stop before
hitting the cab of the lorry.
The car has mass 1250 kg and is moving at a speed of 28.0 m s¯ª. The lorry has mass 3500 kg
and a speed of 25.5 m sª. The length of the flat back of the lorry allows a braking distance of
5.0 m.
By considering both momentum and energy show that the stunt is possible, provided a
minimum force of about 600 N slows the car down. You should support your explanations with
calculations.
Treat the situation as one in which two objects join together.
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