velocity and acceleration vector valued functions written by judith mckaig assistant professor of...
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Velocity and Acceleration
Vector Valued Functions
Written by Judith McKaig Assistant Professor of Mathematics
Tidewater Community College Norfolk, Virginia
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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
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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.
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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
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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
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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
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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
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For comments on this presentation you may email the author Professor Judy Gill [email protected] or the publisher of the VML, Dr. Julia Arnold at [email protected].