Consider a modified Bernoulli equation.

$$
\left(P+\frac{A}{Bt^2} ight)+ ho g(h+Bt)\+\frac{1}{2} ho V^2= ext{constant}
$$

If $𝑡$ has the dimension of time then the dimensions of $A$ and $B$ are ____, ____ respectively.

Question

Consider a modified Bernoulli equation.

$$
\left(P+\frac{A}{Bt^2}ight)+ho g(h+Bt)\+\frac{1}{2}ho V^2=	ext{constant}
$$

If $𝑡$ has the dimension of time then the dimensions of $A$ and $B$ are ____, ____ respectively.

Accepted Answer

B. 1. In the term
   $$
   h+Bt,
   $$
   both quantities being added must have the same dimensions.

Since $h$ represents height,
   $$
   [h]=L.
   $$

Therefore,
   $$
   [B][t]=L.
   $$

As
   $$
   [t]=T,
   $$
   we obtain
   $$
   [B]=LT^{-1}=[M^0LT^{-1}].
   $$

2. In the term
   $$
   P+\frac{A}{Bt^2},
   $$
   the quantity
   $$
   \frac{A}{Bt^2}
   $$
   must have the same dimensions as pressure.

3. Pressure is force per unit area, so
   $$
   [P]=\frac{MLT^{-2}}{L^2}=ML^{-1}T^{-2}.
   $$

4. Since
   $$
   [B]=LT^{-1},
   $$
   we have
   $$
   [Bt^2]=(LT^{-1})(T^2)=LT.
   $$

5. Therefore,
   $$
   \left[\frac{A}{Bt^2}\right]=ML^{-1}T^{-2},
   $$
   which gives
   $$
   [A]=(ML^{-1}T^{-2})(LT)=MT^{-1}.
   $$

Hence,
   $$
   [A]=[ML^0T^{-1}].
   $$

6. Therefore, the required dimensions are
   $$
   [A]=[ML^0T^{-1}],\qquad
   [B]=[M^0LT^{-1}].
   $$

Hence, the correct answer is **Option B**.

Consider a modified Bernoulli equation. (P+ABt2)+ρg(h+Bt)+12ρV2=constant\left(P+\frac{A}{Bt^2}\right)+\rho g(h+Bt)\\+\frac{1}{2}\rho V^2=\text{constant}(P+Bt2A​)+ρg(h+Bt)+21​ρV2=constant If 𝑡𝑡t has the dimension of time then the dimensions of AAA and BBB are ____, ____ respectively.