If a reaction rate decreases by a factor of one-ninth when a reactant concentration is decreased by one-third, what is the order of the reaction with respect to that reactant?

The reaction is second-order in [A].  

For the reaction A ⟶ products

The rate law is r = k[A]^m

Consider two Experiments 1 and 2.

r₂/r₁ = {k[A]₂^m}/{k[A]₁^m }  

r₂/r₁ = {[A]₂/[A]₁}^m

r₂/r₁ = {(⅓[A])/[A]}^m = (⅑r₁)/r₁

(⅓)^m= ⅑

3^m = 9

m = 2

The reaction is second-order in [A].

If dividing the concentration by 3 divides the rate by 9, the reaction is second-order.

d.) circle motion requires a force to give it a centripetal acceleration.

b.final