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Unformatted text preview: er is a vector; the latter a nonnegative real valued function.
Example 2 An object moves on a helical path given by
velocity and acceleration vectors and (2) the speed of the object at
Solution (1)
4 , cos , sin
4, sin , cos
4, 0, sin , cos
cos , sin 4 The speed is given by the norm of the velocity vector:
sin 4
Solution (2) √17 3 cos 4 , cos , sin . Find (1) the
0. The speed of the object at
0 is 0
√17. Interestingly, this trajectory is one of
constant speed. Can we therefore say that the velocity is constant?
Example 3 An object moves on a path given by
sin
1 cos . Sketch its
trajectory. On the trajectory, plot the velocity and acceleration vectors at the point ( , 2 .
Solution sin
1 cos /2 0
0 3
2 1 2 0 3 /2 1 2
2 1
1 0 The trajectory passes through the point ( , 2 when
, that is
, 2 , as can be
seen from the table above. In order to sketch the trajectory, we may attempt to eliminate
the parameter which does not look promising, or graph the functions
and
individually. Regardless of what we do, it is clear that
tends to infinity although it
oscillates in the process. However, since cos  1, 0 1 cos
2. Therefore,
is a bounded function which oscillates between 0 and 2.
The graph below illustrates the trajectory on the interval 0 4: ,2 2.0
1.5
1.0
0.5
0 2 4 6 8 The velocity and acceleration vectors at 2 12 are: sin
1 10 1 cos cos sin 0 Similarly,
sin cos 0
These vectors are vectors in standard form so technically they have initial points at the
origin. However, it is customary when dealing with motion to sketch them with their initial
points at the point on the curve: 4 ,2 2 2.0
1.5
1.0
0.5 1 2 3 4 5 6 More often than not, when we study motion problems we know the forces at work, not the position
vector. The problem is then to reconstruct the position vector from the acceleration vector. We do
this via integration:
Given ; 0
, , , , 0 , , , , , We use the dummy variable because it is good practice in mathematics not to use the same
variable in an equation for two different purposes.
1 2 with the initial conditions
Example 4 Given the acceleration vector
2
0
0,0 , 0
1,2 , determine the velocity and position vectors. What is the object’s
speed at
1?
Solution
0,0 1 2
 2 
0 0 1
1 1,2
1,2 3 1,2 1 3
2 3
The object’s speed at 2
2
2 1:
1 5 3 3 3
2 1
Therefore,
1 4 2
16 4 5.85
The units of...
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This note was uploaded on 10/24/2012 for the course MAC 2313 taught by Professor Lopez during the Spring '10 term at Miami Dade College, Miami.
 Spring '10
 LOPEZ
 Equations

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