# What is the difference between $d|v|/dt$ and $|dv/dt|$ & between $d|x|/dt$ and $|dx/dt|$?

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## Solution 1

$\frac{d|v|}{dt}$ means the rate of change of the magnitude of velocity with respect to time. On the other hand $|\frac{dv}{dt}|$ means the magnitude of the rate of change of velocity with respect to time (i.e. the magnitude of acceleration).
Let's take an example: A body was travelling at 5m/s along the positive x - axis. Its velocity changed to 3m/s along -ve x - axis in 1 second. Here $\frac{d|v|}{dt}$ is $\frac{3-5}{1} = -2m/s^2$ and $|\frac{dv}{dt}|$ is $\frac{-3-5}{1}=-8m/s^2$.

## Solution 2

$v$ is velocity and $|v|$ is speed. So with $|dv/dt|$ you take the derivative of velocity and with $d|v|/dt$ you take it of speed.

Everytime there is a turn, velocity changes (because direction of motion changes) but it's magnitude (the speed) doesn't necessarily change as well.

• A satellite in constant-speed orbit will have a nonzero $|dv/dt|$ but a zero $d|v|/dt$.
• A car speeding up straight ahead has nonzero values of both.
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### Siddharth Garg

Updated on June 29, 2020

• Here, $v$ represents velocity vector and $a$ represents acceleration vector.
• How can $|\frac{dv}{dt}|$ result in a negative value? The bars give the vector magnitude as an absolute value.