velocity and acceleration vector valued functions written by judith mckaig assistant professor of...

Velocity and Acceleration

Vector Valued Functions

Written by Judith McKaig Assistant Professor of Mathematics

Tidewater Community College Norfolk, Virginia

2 2

Velocity ( ) ( ) ( ) ( )

Acceleration ( ) ( ) ( ) ( )

Speed ( ) ( ) ( ) ( )

v t t x t y t

a t t x t y t

v t t x t y t

i j

i j

r

r

r

Definitions of Velocity and Acceleration:

If x and y are twice differentiable functions of t and r is a vector-valued function given by r(t) = x(t)i + y(t)j, then the velocity vector, acceleration vector, and speed at time t are as follows:

The definitions are similar for space functions of the form: r(t) = x(t)i + y(t)j + z(t)k

2 2 2

Velocity ( ) ( ) ( ) ( ) ( )

Acceleration ( ) ( ) ( ) ( ) ( )

Speed ( ) ( ) ( ) ( ) ( )

v t t x t y t z t

a t t x t y t z t

v t t x t y t z t

i j k

i j k

r

r

r

Example 1: The position vector describes the path of an object moving in the xy-plane. a. Sketch a graph of the path.b. Find the velocity, speed, and acceleration of the object at any

time, t.c. Find and sketch the velocity and acceleration vectors at t = 2

2( )t t t r i j

Solution: a. To help sketch the graph of the path, write the following parametric equations:

( )x t t2( )y t t

2y xThe curve can then be represented by the equation with the orientation as shown in the graph.

c. At t = 2, plug into the equations above to get:the velocity vector v(2) = i + 4j, the acceleration vector a(2) = 2j

v(2) = i + 4ja(2) = 2j

To sketch the graph of the velocity vector, start at the initial point (2,4) and move right 1 and up 4 to the terminal point (3,8). Sketch the acceleration similarly.

So the following vector valued functions represent velocity and acceleration and the scalar for speed:v(t) = i + 2tja(t) = 2j

2 2 2Speed 1 (2 ) 1 4t t

2 2

Velocity ( ) ( ) ( ) ( )

Acceleration ( ) ( ) ( ) ( )

Speed ( ) ( ) ( ) ( )

v t t x t y t

a t t x t y t

v t t x t y t

i j

i j

r

r

r

b.

2( )t t t i jr

Example 2: The position vector describes the path of an object moving in the xy-plane. a. Sketch a graph of the path.b. Find the velocity, speed, and acceleration of the object at any

time, t.c. Find and sketch the velocity and acceleration vectors at (3,0)

( ) 3cos 2sint t t r i j

Solution: a. To help sketch the graph of the path, write the following parametric equations:

x3cos , so cos

3

2sin , so sin2

x t t

yy t t

Since , the curve can be

represented by the equation

which is an ellipse with the orientation as shown in the graph.

2 2sin cos 1t t 2 2

19 4

x y

x

y

c. The point (3,0) corresponds to t = 0. You can find this by solving:

3cos t = 3cos t = 1t = 0

At t = 0, the velocity vector is given by v(0) = 2j, and the acceleration vector is given by a(0) = -3i

b. By differentiating each component of the vector, you can find the following vector valued functions which represent velocity and acceleration. You can use the formula to find the scalar for speed:v(t) = -3sinti + 2costja(t) = -3costi-2sintj

2 2

2 2

Speed ( 3sin ) (2cos )

Speed 9sin 4cos

t t

t t

x

y

v(0)=2j

a(0)=-3i

r(t) = 3costi + 2sintj

Example 3: The position vector r describes the path of an object moving in space. Find the velocity, acceleration and speed of the object.

32 2( ) , , 2t t t tr

Solution: Recall, you are given r(t) in component form. It can be written in standard form as:

32 2( ) 2t t t t i j kr

The velocity and acceleration can be found by differentiation:1

2( ) 2 3t t t i j kv1

23

( ) 22

t t

i ka

The speed is found using the formula and simplifying:

2Speed= ( ) 4 9 1t t t v

For comments on this presentation you may email the author Professor Judy Gill atjgill@tcc.edu or the publisher of the VML, Dr. Julia Arnold at jarnold@tcc.edu.