Answer :
The velocity vector is equal to 10 . [tex](\frac{8}{10},0,\frac{6}{10} )[/tex]
Explanation
The particle's velocity is the derivative of its position. The particle's acceleration is the derivative of its velocity. You can calculate the velocity and acceleration as follows:
[tex]\vec{v}(t)=\frac{d r}{d t}=\frac{d}{d t}(8 t+5) \vec{i}+\frac{d}{d t}\left(8 t^2-2\right) \vec{j}+\frac{d}{d t}(6 t) \vec{k}=8 \vec{\imath}+16 t \vec{j}+6 \vec{k}[/tex]
[tex]\vec{a}(t)=\frac{d v}{d t}=\frac{d}{d t}(8) \vec{i}+\frac{d}{d t}(16 t) \vec{j}+\frac{d}{d t}(6) \vec{k}=0 \vec{i}+16 \vec{j}+0 k[/tex]
The velocity at t = 0 is [tex]\vec{v}(0)=8 \vec{i}+0 \vec{j}+6 k[/tex]
if the speed at t = 0 is [tex]||\vec{v}(0)||[/tex] = 10 then the velocity at t = 0 can be written as a product of the speed at t = 0 & the direction at t = 0 is
[tex]\vec{v}(0)=\|\vec{v}(0)\| \frac{\vec{v}(0)}{\|\vec{v}(0)\|}=10 \cdot\left(\frac{8}{10}, 0, \frac{6}{10}\right)[/tex]
What is velocity?
Velocity is the directional speed of a moving item as an indication of its rate of change in position as seen from a specific frame of reference and measured by a specific time standard (e.g., 60 km/h northbound). The idea of velocity is crucial in kinematics, the part of classical mechanics that explains the motion of bodies.
A physical vector quantity called velocity requires both magnitude and direction to be defined. Speed is a coherent derived unit that expresses the scalar absolute value (magnitude) of velocity. Its quantity is expressed in SI units (metric system) of metres per second (m/s or ms1). For example, "5 meters per second" is a scalar while "5 meters per second east" is a vector.
Learn more about Velocity
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Full question
The equation r (t) = (8t + 5)i + (8t^2 - 2)j + (6t )k is the position of a particle in space at time t. Find the particle's velocity and acceleration vectors. Then write the particle's velocity at t equals 0 as a product of its speed and direction. What is the velocity vector?
