Question

1.In the equation (𝑃+𝑎/𝑉²)(𝑉−𝑏)=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, determine the units of a and b.

2. The square of the speed v of an object undergoing acceleration a is a variable of a function a and the displacement S according to the expression given by

𝑣²=𝑘𝑎^𝑚𝑆^𝑛

What dimensions should k have in order for the expression to be dimenssionally consistent?

Answer 1

1)

[b]=[L^3]=m^3\\ [a]=Pa(m^3)^2=Nm^4

2)

𝑣^2=𝑘𝑎^𝑚𝑆^𝑛\\ [𝑣]^2=[𝑎]^𝑚[𝑆]^𝑛\\ [LT^{-1}]^2=[LT^{-2}]^𝑚[L]^𝑛\\ -2=-2m\to m=1\\ 2=m+n=1+n\to n=1

k will be dimensionless constant.