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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.
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.)
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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