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The three-dimensional motion of a particle is defined by the position
Fig. P11.96
(a)

The magnitude of the velocity (v) and acceleration (a) when time is 0 sec.
Answer to Problem 11.96P
The magnitude of the velocity (v) and acceleration (a) when time is 0 sec are
Explanation of Solution
Given Information:
The three dimensional motion of a particle is defined by the position vector is
The curve described by the particle lies on the hyperboloid is
The value of A and B are 3 and 1 respectively.
Calculation:
Write the three dimensional motion of a particle position vector equation.
Here, x is
Consider x:
Consider y:
Consider z:
Calculate the
Substitute
Check whether the position vector equation satisfied the curve equation or not.
Substitute
Hence, the equation is satisfied.
Rewrite the Equation (1).
Substitute 3 for A and 1 for B in Equation (1).
Write the expression for velocity using the relation:
Substitute
Calculate velocity vector
Substitute 0 for t in Equation (5).
Here,
Calculate the magnitude of velocity (v) using the relation:
Substitute
Write the expression for acceleration vector using the relation:
Substitute
Substitute 0 sec for t.
Here,
Calculate the magnitude
Substitute 0 is
Therefore, the magnitude of the velocity (v) and acceleration (a) when time is 0 sec are
(b)

The smallest nonzero value of t for which the position vector and the velocity are perpendicular to each other.
Answer to Problem 11.96P
The smallest nonzero value of t for which the position vector and the velocity are perpendicular to each other is
Explanation of Solution
Given Information:
The three dimensional motion of a particle is defined by the position vector is
The curve described by the particle lies on the hyperboloid is
The value of A and B are 3 and 1 respectively.
Calculation:
Write the equation if the position vector and velocity vector are perpendicular:
Substitute
Using trial and error method the smallest root is (t) is 4.38 sec.
Therefore, the smallest nonzero value of t for which the position vector and the velocity are perpendicular to each other is
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