Question

Which of the following are dimensionally consistent? (Choose all that apply.)(a) a=v / t+xv2 / 2(b) x=3vt(c) xa2=x2v / t4(d) x=vt+vt2 / 2(e) v=x2 / at3(f) a3=x2v / t5(g) x=t(h) v=5at

Answer 1

Complete Question

The  complete question is shown on the first uploaded image

Answer:

A

is dimensionally consistent

B

is not dimensionally consistent

C

is dimensionally consistent

D

is not dimensionally consistent

E

is not dimensionally consistent

F

is dimensionally consistent

G

is dimensionally consistent

H

is not dimensionally consistent

Step-by-step explanation:

From the question we are told that

   The equation are

                        $A) \   \  a^3  =  \frac{x^2 v}{t^5}$

                       

                       $B) \   \  x  =  t$

 

                       $C \ \ \ v  =  \frac{x^2}{at^3}$

 

                      $D \ \ \ xa^2 = \frac{x^2v}{t^4}$

                      $E \ \ \ x  = vt+ \frac{vt^2}{2}$

                     $F \ \ \  x = 3vt$

 

                    $G \ \ \  v =  5at$

 

                    $H \ \ \  a  =  \frac{v}{t} + \frac{xv^2}{2}$

Generally in dimension

     x - length is represented as  L

     t -  time is represented as T

     m = mass is represented as M

Considering A

           $a^3  =  (\frac{L}{T^2} )^3 =  L^3\cdot T^{-6}$

and    $\frac{x^2v}{t^5 } =  \frac{L^2 L T^{-1}}{T^5}  =  L^3 \cdot T^{-6}$

Hence

           $a^3  =  \frac{x^2 v}{t^5}$ is dimensionally consistent

Considering B

            $x =  L$

and      

            $t = T$

Hence

      $x  =  t$  is not dimensionally consistent

Considering C

     $v  =  LT^{-1}$

and  

    $\frac{x^2 }{at^3} =  \frac{L^2}{LT^{-2} T^{3}}  =  LT^{-1}$

Hence

   $v  =  \frac{x^2}{at^3}$  is dimensionally consistent

Considering D

    $xa^2  = L(LT^{-2})^2 =  L^3T^{-4}$

and

     $\frac{x^2v}{t^4}  = \frac{L^2(LT^{-1})}{ T^5} =  L^3 T^{-5}$

Hence

    $xa^2 = \frac{x^2v}{t^4}$  is not dimensionally consistent

Considering E

   $x =  L$

;

   $vt  =  LT^{-1} T =  L$

and  

    $\frac{vt^2}{2}  =  LT^{-1}T^{2} =  LT$

Hence

   $E \ \ \ x  = vt+ \frac{vt^2}{2}$   is not dimensionally consistent

Considering F

     $x =  L$

and

    $3vt = LT^{-1}T =  L$      Note in dimensional analysis numbers are

                                                       not considered

  Hence

       $F \ \ \  x = 3vt$  is dimensionally consistent

Considering G

    $v  =  LT^{-1}$

and

    $at =  LT^{-2}T =  LT^{-1}$

Hence

      $G \ \ \  v =  5at$   is dimensionally consistent

Considering H

     $a =  LT^{-2}$

,

       $\frac{v}{t}  =  \frac{LT^{-1}}{T}  =  LT^{-2}$

and

    $\frac{xv^2}{2} =  L(LT^{-1})^2 =  L^3T^{-2}$

Hence

    $H \ \ \  a  =  \frac{v}{t} + \frac{xv^2}{2}$  is not dimensionally consistent

Answer 2

We want to see which ones of the given expressions are dimensionally consistent. We will see that the correct options are:

• a) x = 3*v*t
• h) v = 5*a*t

What means to be dimensionally consistent?

This means that we have the same units in the left and in the right side of the equation.

The units are:

• a = [m/s^2]
• x = [m]
• v = [m/s]
• t = [s]

Now we can analyze the expressions to see the units in each one, I will show you how to do it:

a) a = v/t + x*v^2

Replacing the units we have:

[m/s^2] = [m/s]/[s] + [m]*[m^2/s^2]

[m/s^2] = [m/s^2] + [m^3/s^2]

You can see that we have an m^3 in the right side, so these are not equivalent.

b) x = 3*v*t

Replacing the units we have:

[m] = 3*[m/s]*[s] = 3*[m]

So yes, the units are the same in both sides, so this is dimensionally consistent.

With the same procedure we can see that:

• c) [m^3/s^2] = [m^3/s]                not consistent
• d) [m] = [m] + [m*s]                     not consistent
• e) [m/s] = [m^2]                           not consistent
• f) [m^3/s^6] = [m^3/s]                 not consistent
• g) [m] = [s]                                   not consistent
• h) [m/s] = 5*[m/s]                        consistent

So the correct options are b and h.

If you want to learn more about dimensions, you can read:

https://brainly.com/question/20384972