A force defined by F = αt² + βt acts on a particle at a given time t. The factor which is dimensionless, if α and β are constants, is :

We are given the force as a function of time:

$F=\alpha t^2+\beta t$

To find a dimensionless factor involving the constants α and β, we first determine the dimensions of each constant by requiring each term to have the dimensions of force.

The dimension of force is:

$\mathrm{[F]}=M\times L\times T^{-2}$

Since time t has dimension T, the first term α t² must have the same dimensions as F:

$\mathrm{[\alpha ]}\times T^2=M\times L\times T^{-2}$

Therefore:

$\mathrm{[\alpha ]}=M\times L\times T^{-4}$

Similarly, for the second term β t:

$\mathrm{[\beta ]}\times T=M\times L\times T^{-2}$

Thus:

$\mathrm{[\beta ]}=M\times L\times T^{-3}$

Now we seek a combination of α, β and t that is dimensionless. Consider the product α t:

$\mathrm{[\alpha \; t]}=M\times L\times T^{-4}\times T=M\times L\times T^{-3}$

This has exactly the same dimensions as β. Consequently, the ratio (α t) / β is dimensionless:

$\frac{\alpha t}{\beta }$

Therefore the required dimensionless factor is α t / β.