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I read in the book Concepts of Physics by HC Verma in the section of Limitations of Dimensional analysis that the method of dimensions cannot lead us to the correct expression sometimes if expression is not of product type. What does "an expression of product type" mean in this context?

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An expression of product type is one that only involves multiplication (which includes division and exponentiation), but no addition (or subtraction).

For example, we can use dimensional analysis to deduce that the function for displacement $s$ in terms of constant acceleration $a$ and time $t$, with zero initial velocity, must be of the form $$s = kat^2$$ for some dimensionless constant $k$. The expression $kat^2$ is of product type because it only involves multiplications.

If the initial velocity is not zero but is instead $u$ we get

$$s = jut + kat^2$$ where $j$ is again some dimensionless constant. We can add $jut$ to $kat^2$ because they have the same dimensions. But now our expression for $a$ contains an addition, so it is not of product type, and simple dimensional analysis does not give us any relationship between the values of these 2 constants. Luckily, we can determine them using calculus. ;)

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