Answer :
The equation of motion in terms of v and x is [tex]\frac{dv}{dx} = -\mu v[/tex]
Explanation
Let [tex]v(t) = \frac{ds}{dt}[/tex] be the velocity of a time t . Now its time rate of change, [tex]\acute{v}[/tex] = [tex]\frac{dv}{dt}[/tex]
is the acceleration, Therefore,
⇒ [tex]a(t)=v^{\prime}(t)=\frac{d}{d t}\left(\frac{d s}{d t}\right)[/tex]
On the other hand, the acceleration will be given by
⇒ [tex]a = - \mu v^2[/tex]
Where [tex]\mu[/tex] is constant, thus
⇒ [tex]\frac{dv}{dt} = -\mu v^2[/tex]
Where is the motion equation expressed in terms of v and t We must now write it in terms of v and x. To accomplish this, we make use of the fact that
⇒ [tex]\frac{d v}{d t}=\frac{d v}{d x} \cdot \frac{d x}{d t}=v \frac{d v}{d x}[/tex]
We now substitute the acquired result into the supplied equation.
⇒ [tex]v \frac{d v}{d x}=\frac{d v}{d t}=-\mu v^2[/tex]
Divide the equation by v ≠ 0
⇒ [tex]\frac{dv}{dx} = -\mu v[/tex]
We have changed the provided equation of motion in terms of v and x in this manner.
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
A rocket sled having an initial speed of 160 mi/h is slowed by a channel of water. Assume that, during the braking process, the acceleration a is given by
[tex]av = -\mu v^2[/tex]
where v is the velocity and \mu is a constant. (a) As in Example 4 in the text, use the relation
[tex]\frac{dv}{dt} = v\frac{dv}{dx}[/tex]
to write the equation of motion in terms of v and x.
