In the absence if air resistance, it is plausible that thetime, t, that it takes

Question

In the absence if air resistance, it is plausible that thetime, t, that it takes for an object to fall (starting from rest)to the ground is a function of its initial height, h, and theacceleration of gravity, g. assume that the relation between t, h,and g has the following form:       t = C hagb where C, a, and b are (as of yet) unknown dimensionlessconstants. find values for the constants a and b that will makeboth sides of this equation have the same dimensions. can we findthe value of C using this method? In the absence if air resistance, it is plausible that thetime, t, that it takes for an object to fall (starting from rest)to the ground is a function of its initial height, h, and theacceleration of gravity, g. assume that the relation between t, h,and g has the following form:       t = C hagb where C, a, and b are (as of yet) unknown dimensionlessconstants. find values for the constants a and b that will makeboth sides of this equation have the same dimensions. can we findthe value of C using this method? where C, a, and b are (as of yet) unknown dimensionlessconstants. find values for the constants a and b that will makeboth sides of this equation have the same dimensions. can we findthe value of C using this method?

Explanation / Answer

In the absence if air resistance, it is plausible that thetime, t, that it takes for an object to fall (starting from rest)to the ground is a function of its initial height, h, and theacceleration of gravity, g. assume that the relation between t, h,and g has the following form       t = C hagb where C, a, and b are (as of yet) unknown dimensionlessconstants. find values for the constants a and b that will makeboth sides of this equation have the same dimensions.
t=C ha gb
writing dimensional formulas oneither side

[T]=C{L]a[LT-2]b
simplifying the given equation
[T]=C{LaLbT-2b]=C[La+bT-2b]

this can also be written as
[LoT]=C[La+bT-2b]
according to homogenity of equation similar terms on eitherside of the equation should have the dimension
hence comparing the powers of L and T on either side of theequation
we have
a+b=0
and
-2b=1
so b=-1/2 a= 1/2
can we find the value of C using this method?
NO

where C, a, and b are (as of yet) unknown dimensionlessconstants. find values for the constants a and b that will makeboth sides of this equation have the same dimensions.
t=C ha gb
writing dimensional formulas oneither side

[T]=C{L]a[LT-2]b
simplifying the given equation
[T]=C{LaLbT-2b]=C[La+bT-2b]

this can also be written as
[LoT]=C[La+bT-2b]
according to homogenity of equation similar terms on eitherside of the equation should have the dimension
hence comparing the powers of L and T on either side of theequation
we have
a+b=0
and
-2b=1
so b=-1/2 a= 1/2
can we find the value of C using this method?
NO

Related Questions

Navigate

• Browse (All)
• Browse I
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.