Question

What is the numerical value of ratio of the instantaneous velocity to instantaneous speed?
A. Always less than 1
B. Always equal to 1
C. Always more than 1
D. Equal to or less than 1

Answer 1

The expression for instantaneous velocity is
$v\left( t \right) = \mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{x\left( {t + \Delta t} \right) - x\left( t \right)}}{{\Delta t}} = \dfrac{{dx\left( t \right)}}{{dt}}$
The expression for instantaneous speed is
Instantaneous speed $= \left| {v\left( t \right)} \right|$
The magnitudes of speed and velocity are equal at an instant.

Complete step-by-step solution :Instantaneous velocity is the quantity which tells the speed of a moving object in its path. It is also called simple velocity. This is the average velocity of the two points along the path in the limit at the time between two points becomes zero. It is the derivative of $x$with respect to $t$.
$v\left( t \right) = \mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{x\left( {t + \Delta t} \right) - x\left( t \right)}}{{\Delta t}} = \dfrac{{dx\left( t \right)}}{{dt}}$
We can find the average speed by this expression, Average Speed = $s = \dfrac{{T.D}}{{E.T}}$
where, $T.D =$Total Distance
$E.T =$Elapsed Time
To get the instantaneous speed from the magnitude of instantaneous velocity
Instantaneous speed $= \left| {v\left( t \right)} \right|$
Ratio of instantaneous velocity to instantaneous speed is
$\dfrac{v}{s} = \dfrac{{\xrightarrow{{{v_{ins}}}}}}{{\left| {{V_{ins}}} \right|}}$
where, $v =$instantaneous velocity
$s =$instantaneous speed
$\because$At an instant, the magnitudes of speed and velocity will be equal
$\therefore \dfrac{v}{s} = \dfrac{{\xrightarrow{{{v_{ins}}}}}}{{\left| {{V_{ins}}} \right|}} = 1$
Therefore, option (A) always equal to 1 is the correct answer.

Note:- The instantaneous velocity of an object is the limit of the average velocity as the elapsed time approaches zero, or the derivative of $x$with respect to $t.$
$v\left( t \right) = \dfrac{d}{{dt}}x\left( t \right)$