Question

Which of the following relations are always true?
$\overrightarrow{v}=velocity$, $\overrightarrow{a}=$acceleration, $K=\frac{1}{2}m{v}^2=$ kinetic energy:

• A. $\frac{dK}{dt}=m\overrightarrow{v}.\overrightarrow{a}$
• B. $\frac{d\left|\overrightarrow{v}\right|}{dt}=\frac{\overrightarrow{a}.\overrightarrow{v}}{\left|\overrightarrow{v}\right|}$
• C. $\frac{d\left|\overrightarrow{v}\right|}{dt}=\left|\frac{d\overrightarrow{a}}{dt}\right|$
• D. $\Delta \left|\overrightarrow{a}\right|=\left|{\int }_{t1}^{t2}\overrightarrow{a}dt\right|$

Answer 1

The correct option is B $\frac{d\left|\overrightarrow{v}\right|}{dt}=\frac{\overrightarrow{a}.\overrightarrow{v}}{\left|\overrightarrow{v}\right|}$

For option $\left(A\right)$

$K=\frac{1}{2}m{v}^2$

$\frac{dK}{dt}=\frac{1}{2}m\left(2v\right)\frac{dv}{dt}$

$=mva=m\left(\overrightarrow{v}.\overrightarrow{a}\right)$

Hence, this option is correct.

For option $\left(B\right)$

$\frac{d\left|\overrightarrow{v}\right|}{dt}=\frac{\overrightarrow{a}.\overrightarrow{v}}{\left|\overrightarrow{v}\right|}=\frac{\left|\overrightarrow{a}\right|\left|\overrightarrow{v}\right|\cos \left(\overrightarrow{a}\overrightarrow{v}\right)}{\left|\overrightarrow{v}\right|}$ [cancle $\left|\overrightarrow{v}\right|$ both above and below]

so, this option is also true because $\frac{d\left|\overrightarrow{v}\right|}{dt}$ means change in magnitude of velocity which is that acceleration which is in the direction of velocity and $\frac{d\left|\overrightarrow{v}\right|}{dt}$ also meant by acceleration

Which is in the direction of velocity.

For option $\left(C\right)$

$\frac{d\left|\overrightarrow{v}\right|}{dt}\ne \left|\frac{d\overrightarrow{a}}{dt}\right|$

This option is false because in magnitude of velocity may be negative because of retardation.

For option $\left(D\right)$

$\Delta \left|\overrightarrow{a}\right|=\left|{\int }_{t1}^{t2}\overrightarrow{a}dt\right|$

this option is false because L.H.S is meant for change in magnitude of acceleration and R.H.S meant for area of $a-t$ graph