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Answer
At time
, the velocity component in the direction
is
and the acceleration component is
.
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To find the components of the velocity and acceleration of the particle at time
, we first need to compute the velocity vector
by differentiating the position functions
,
, and
with respect to time
.
The velocity components are given by:
At
:
Thus, the velocity vector is
.
Next, we compute the acceleration vector
by differentiating the velocity components:
This gives us the acceleration vector as
.
To find the components of both vectors in the direction of
, we first need to normalize this direction vector:
The unit direction vector
is then:
Now, to project the velocity and acceleration onto this direction, we use the dot product:
Thus, the final results for the components of the velocity and acceleration at
in the specified direction are: