Kinematics exercise got the best of me.

Question

The position of a particle moving along the x axis varies in time according to the expression x = 3t^2 , where x is in meters and t is in seconds. Evaluate its position

(a) at t = 3.00 s

(b) at 3.00 s + ∆t

(c) Evaluate the limit of ∆x/∆t as ∆t approaches zero to find the velocity at t = 3.00 s.

Answer 1

(a)

x(3) = 3×3² = 27

(b)

x(3 + Δt) = 3×(3 + Δt)² = 3×(9 + 6Δt + (Δt)²) = 27 + 18Δt + 3(Δt)²

(c)

It is important to know that

Δx = x(3 + Δt) - x(3)

Then

lim(Δx/Δt) =

lim((18Δt + 3(Δt)²)/Δt) =

lim(18 + 3Δt) = 18

So the velocity at time t = 3.00 is 18 m/s.