Question

An object moves with constant acceleration a. Which of the following expressions are also constant?

• A. $\frac{d|v|}{dt}$
• B. $|\frac{dv}{dt}|$
• C. $\frac{d\left(v^2\right)}{dt}$
• D. $\frac{d\left(\frac{v}{|v|}\right)}{dt}$

Answer 1

The correct option is A $\frac{d|v|}{dt}$

Given,

Velocity vector is $\overrightarrow{v}\left(t\right)\text{=}\left|\overrightarrow{v}\right|\hat{v}$

If acceleration is constant, velocity uniformly change with time.

Acceleration vector $\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}$

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

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

If acceleration is constant, whole the time, unit direction of velocity & unit direction of acceleration is same.

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

if acceleration is constant then $\left|\overrightarrow{a}\right|$ is constant.

Hence, $\frac{d\left|\overrightarrow{v}\right|}{dt}$ is constant.

Alternatively, the modulus of velocity vector $|v|$ will be constant and derivative of a constant, $\frac{d|v|}{dt}$ is zero or a constant.