Question

Position of a body with acceleration 'a' is given by x=Ka^m t^n, here t is time. Find value of m and n​

Answer 1

Assuming that K is a dimensionless constant,

we have,
$x = k {a}^{m} {t}^{n}$

For x is displacement,

Therefore it's dimensions must be,
${M}^{0} {L}^{1} {T}^{0}$



Similarly,
On the RHS,
putting the dimensions into the picture,

${({M}^{0} {L}^{1} {T}^{ - 2}) }^{m} { ({T}^{1}) }^{n} \\ \\ {L}^{m} {T}^{ - 2m} ( {T}^{n}) \\ \\ {L}^{m} {T}^{n - 2m}$

Since the units of LHS and RHS must be same for this equation to be valid,

equating the powers,

m = 1
and


n-2m = 0

Using value of m here,

n- 2 = 0
n = 2


Therefore m= 1 and n = 2.