Exercises 1 [de Mestre] Consider a projectile, launched with initial velocity vo, at an angle 0. To study its motion we may guess that these are the relevant quantities. dimensional quantity horizontal position x vertical position y initial speed vo angle of launch 0 formula L'M°T° L'M°T° L'MºT-1 LOMOTO acceleration due to gravity g L'MºT-2 time t LOM°T¹ (a) Show that {gt/vo, gx/v2, gy/v2, 0} is a complete set of dimensionless products. (Hint. One way to go is to find the appropriate free variables in the linear system that arises but there is a shortcut that uses the properties of a basis.)
Exercises 1 [de Mestre] Consider a projectile, launched with initial velocity vo, at an angle 0. To study its motion we may guess that these are the relevant quantities. dimensional quantity horizontal position x vertical position y initial speed vo angle of launch 0 formula L'M°T° L'M°T° L'MºT-1 LOMOTO acceleration due to gravity g L'MºT-2 time t LOM°T¹ (a) Show that {gt/vo, gx/v2, gy/v2, 0} is a complete set of dimensionless products. (Hint. One way to go is to find the appropriate free variables in the linear system that arises but there is a shortcut that uses the properties of a basis.)
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Please do Exercise 1 part A and B and please show step by step and explain
![(b) These two equations of motion for projectiles are familiar: x= vo cos(0)t and
y = vo sin(0)t (g/2)t². Manipulate each to rewrite it as a relationship among
the dimensionless products of the prior item.
2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might
be determined by the same forces between atoms as determine the solid's ordinary
elastic behavior. The relevant quantities are these.
quantity
characteristic frequency v
compressibility k
dimensional
formula
L°M°T-1
L¹M-¹T²
L-³ M°T°
number of atoms per cubic cm N
mass of an atom m
LM¹Tº
Show that there is one dimensionless product. Conclude that, in any complete
relationship among quantities with these dimensional formulas, k is a constant
times v-2N-1/³m-¹. This conclusion played an important role in the early study
of quantum phenomena.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F9b8a26ba-2213-4a5e-bf53-85abda970e9f%2Fqed3a9i_processed.png&w=3840&q=75)
Transcribed Image Text:(b) These two equations of motion for projectiles are familiar: x= vo cos(0)t and
y = vo sin(0)t (g/2)t². Manipulate each to rewrite it as a relationship among
the dimensionless products of the prior item.
2 [Einstein] conjectured that the infrared characteristic frequencies of a solid might
be determined by the same forces between atoms as determine the solid's ordinary
elastic behavior. The relevant quantities are these.
quantity
characteristic frequency v
compressibility k
dimensional
formula
L°M°T-1
L¹M-¹T²
L-³ M°T°
number of atoms per cubic cm N
mass of an atom m
LM¹Tº
Show that there is one dimensionless product. Conclude that, in any complete
relationship among quantities with these dimensional formulas, k is a constant
times v-2N-1/³m-¹. This conclusion played an important role in the early study
of quantum phenomena.
![Exercises
1 [de Mestre] Consider a projectile, launched with initial velocity vo, at an angle 0.
To study its motion we may guess that these are the relevant quantities.
dimensional
quantity
horizontal position x
vertical position y
initial speed vo
angle of launch 0
formula
L¹M°T°
L'M°T°
L'MºT-1
LOMOTO
acceleration due to gravity g
L'MºT-2
time t
LºM°T¹
(a) Show that {gt/vo, gx/v2, gy/v2, 0} is a complete set of dimensionless products.
(Hint. One way to go is to find the appropriate free variables in the linear system
that arises but there is a shortcut that uses the properties of a basis.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F9b8a26ba-2213-4a5e-bf53-85abda970e9f%2Ffpe76zo_processed.png&w=3840&q=75)
Transcribed Image Text:Exercises
1 [de Mestre] Consider a projectile, launched with initial velocity vo, at an angle 0.
To study its motion we may guess that these are the relevant quantities.
dimensional
quantity
horizontal position x
vertical position y
initial speed vo
angle of launch 0
formula
L¹M°T°
L'M°T°
L'MºT-1
LOMOTO
acceleration due to gravity g
L'MºT-2
time t
LºM°T¹
(a) Show that {gt/vo, gx/v2, gy/v2, 0} is a complete set of dimensionless products.
(Hint. One way to go is to find the appropriate free variables in the linear system
that arises but there is a shortcut that uses the properties of a basis.)
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