Physics

Problem

The velocity

v

attained from rest under uniform acceleration

a

acting for time

t

is assumed to obey

v = k a^{m} t^{n}

, where

k

is a dimensionless constant. Using dimensional reasoning, find the values of

m

and

n

. a)

m=\tfrac{1}{2},\;n=\tfrac{1}{2}

m=1,\;n=1

m=1,\;n=-1

m=1,\;n=2

Intuitive explanation

Dimensional analysis lets you find unknown exponents by requiring that both sides of an equation have matching dimensions. Here, you know $v$ has dimensions of $[LT^{-1}]$ , while $a$ is $[LT^{-2}]$ and $t$ is $[T]$ . Write out the dimensions on both sides, set the exponents of each dimension equal, and solve the resulting system—this forces $m$ and $n$ to have specific values that make the equation dimensionally consistent.

Intuitive explanation

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