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.