Question

Suppose quantity s is a length and quantity t is a time. Suppose the quantities vand aare defined by v = ds/dt and a = dv/dt. (a) What is the dimension of v? (b) What is the dimension of the quantity a?
What are the dimensions of (c)vdt, (d) a dt, and (e) da/dt?

Answer 1

Explanation:

(a) Velocity is given by :

$v=\dfrac{ds}{dt}$

s is the length of the distance

t is the time

The dimension of v will be, $[v]=[LT^-1]$      

(b) The acceleration is given by :

$a=\dfrac{dv}{dt}$

v is the velocity

t is the time

The dimension of a will be, $[a]=[LT^{-2}]$

(c) Since, $d=\int\limits{v{\cdot}dt} =[LT^{-1}][T]=[L]$

(d) Since, $v=\int\limits{a{\cdot}dt} =[LT^{-2}][T]=[LT^{-1}]$

(e)

$\dfrac{da}{dt}=\dfrac{[LT^{-2}]}{[T]}$

$\dfrac{da}{dt}=[LT^{-3}]}$

Hence, this is the required solution.