Physical Derivatives (Practice)

Problem 1

Find the specified parametrization of the first quadrant part of the circle .

(a). In terms of the Coordinate

(b). In terms of the Angle between the Tangent line and the Positive -axis

(c). In terms of the Arc Length from

Problem 2

Consider the following time-parametrized curve:

List the three points , , and in chronological order.

Problem 3

At what points in the -plane does the curve cross itself? What is the difference in between the first time the curve crosses through a point, and the last?

The first intersection occurs at where the difference in is and the difference being .

Problem 4

A circle of radius rolls along the -axis in the positive direction, starting with its centre at . In that position, we mark the topmost point on the circle . As the circle moves, moves with it. Let be the angle the circle has rolled.

(a). Give the Position of the Centre of the Circle as a Function of

(b). Give the Position of as a Function of

Problem 5

A particle traces out a curve in space, so that its position at time is

Let the positive -axis point vertically upwards. When is the particle moving upwards, and when is it moving downwards? Is it moving faster at time or at time ?

The particle is travelling faster at than at

Problem 7

What is the relationship between velocity and speed in a vector-valued function of time?

The speed is the vector magnitude of the velocity.

Problem 8

Let be a vector-valued function. Let , , and denote derivatives with respect to .

Express

in terms of , , , and .

[ ] A.
[ ] B.
[x] C.
[ ] D. 0
[ ] E. None of the above

Problem 9

Show that, if the position and velocity vectors of a moving particle are always perpendicular, then the path of the particle lies on a sphere.

The distance from the centre is always a constant so it must always be some fixed radius from the centre.

Problem 10

Find the speed of a particle with the given position function

Select the correct answer.

[ ] A.
[ ] B.
[ ] C.
[x] D.
[ ] E.

Problem 11

Find the velocity, speed, and acceleration at time of the particle whose position is

Describe the path of the particle.

The arc of the path is a helix oriented around the positive z-axis.

Problem 12

Let