If F is given by F = (α/β) (1−e−βt2/m), where F is force and m & t are mass and time, then dimension of α will be :
Since, F=αβ⎛⎜⎝1−e−βt2m⎞⎟⎠
Since exponential is an dimensionless quantity
Hence,
−βt2m is a dimensionless quantity
∴ [βt2m]=[M0L0T0]
⇒β[T2M]=[M0L0T0]
Hence, β=[M0L0T0][M][T2]
and the two quantities cannot subtracted added until they don't have same dimension
[F]=αβ
since, [F]=[M1L1T−2]
[M1L1T−2]=[α][M1L0T−2]
∴[α]=[M1L1T−2][M1L0T−2]
[α]=[M2L1T−4]