Question

A particle moves along the plane curve C described by r(t)=ti+t2j. Find the curvature of the plane curve at t=2. Round your answer to two decimal places.

Answer 1

The curvature of the plan curve at t=2 is 0.

Explanation:

First, we find take a derivative of r(t) = ti + t^2j

r'(t) = i + 2tj

we can also write it like r'(t) = < 1, 2t>

Now, we take a magnitude of it

|r'(t)| = √(1)^2 + (2t)^2

|r'(t)| = √17

Now, we assume a variable T(t)

T(t) = r'(t)/|r'(t)|

T(t) = (i + 2tj) / √17

T(t) = 1/√17 (< 1, 2t>)

Take derivative of T(t)

T'(t) = 1/√17 (< 0, 2>)

|T'(t)| = √(1/√17)^2 + (0)^2 + (2)^2

|T'(t)| = √69/17

Therefore,

Curvature = |T'(t)| / |r'(t)|

Curvature = (√69/17) / √17

Curvature = √69 .