An object moves velocity v having constant accelaration a- Which of the following expressions are variable quantities -

Let nbsp;v=v_xî+v_yĵ+v_zk̂ nbsp; and a=a_xî+a_yĵ+a_zk̂=dv_xdtî+dv_ydtĵ+dv_zdtk̂ As nbsp;a nbsp;is constant, so its magnitude as well as direction is not ...

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Standard XII

Physics

Introduction to acceleration

An object mov...

Question

An object moves velocity $\to v$ having constant accelaration $\to a$ . Which of the following expressions are variable quantities ?

\frac{d | \to v |}{d t}

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∣ ∣ ∣ ∣ \frac{d \to v}{d t} ∣ ∣ ∣ ∣

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\frac{d (| \to v |^{2})}{d t}

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\frac{d ⎛ ⎜ ⎝ \frac{\to v}{- \to | v |} ⎞ ⎟ ⎠}{d t}

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Solution

The correct option is D $\frac{d ⎛ ⎜ ⎝ \frac{\to v}{- \to | v |} ⎞ ⎟ ⎠}{d t}$
let $\to v = v_{x}^i + v_{y}^j + v_{z}^k$ and
$\to a = a_{x}^i + a_{y}^j + a_{z}^k = \frac{d v_{x}}{d t}^i + \frac{d v_{y}}{d t}^j + \frac{d v_{z}}{d t}^k$
As $\to a$ is constant, so its magnitude as well as direction is not changing, but $v_{x}, v_{y}$ and $v_{z}$ all can be varying (any combination of these if $a_{x}, a_{y} or a_{z}$ , then corresponding velocity component may be constant.)

$A)$
$| \to v | = \sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}$
Hence $\frac{d | \to v |}{d t}$ is variable.
$∴$ Option(a) is correct.

$B)$
By definition
$\frac{d \to v}{d t} = \to a$
hence $\frac{d \to v}{d t}$ is constant

$∴$ Option(b) is incorrect.

$C)$
$| \to v |^{2} = v_{x}^{2} + v_{y}^{2} + v_{z}^{2}$
$\frac{d | \to v |^{2}}{d t}$ is variable
$∴$ Option(c) is correct.

$D)$
$\frac{d ⎛ ⎜ ⎝ \frac{\to v}{- \to | v |} ⎞ ⎟ ⎠}{d t} = \frac{d ⎛ ⎜ ⎜ ⎝ \frac{v_{x}^i + v_{y}^j + v_{z}^k}{\sqrt{v_{x}^{2} + v_{y}^{2} + v_{z}^{2}}} ⎞ ⎟ ⎟ ⎠}{d t}$ is variable quantity.

$∴$ Option(d) is correct.

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An object moves velocity →v having constant accelaration →a. Which of the following expressions are variable quantities ?

An object moves velocity $\to v$ having constant accelaration $\to a$ . Which of the following expressions are variable quantities ?