Question

Find T (t), N (t), aT, and aN at the given time t for the space curve r (t). [Hint: Find a (t), T (t), aT, and aN . Slove for N in the equation a (t) = aT T + aN N.]

Function                                                               Time

Answer 1

Step 1 :

If is the position vector of for a smooth curve C, then the tangential and normal components of acceleration are as follows.

and .

The unit tangent vector is .

Step 2 :

The acceleration is

The position vector is  and time .

Apply derivative on each side with respect to t.

Substitute in above equation.

Find the tangent vector :

Substitute corresponding values in above equation.

Step 3 :

Differentiate with respect to t.

Substitute in above equation.

Find the tangential component of acceleration :

Substitute corresponding values in above equation.

Step 4 :

The normal component of acceleration is :

Substitute corresponding values in above equation.

Find :

The acceleration is

Substitute corresponding values in above equation.

Solution :