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

Options

A. `(d|vecv|)/(dt)`
B. `|(dvecv)/(dt)|`
C. `(d(v^2))/(dt)`
D. `(d(vecv//|vecv|))/(dt)`

Answer

B

Explanation

b. Let `veca = a_xhati + a_yhatj + a_zhatk = (dv_x)/(dt)hati + (dv_y)/(dt)harj + (dv_z)/(dt)hatk`
As `veca` 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`= 0 , then corresponding velocity component may be constant.)
`|vecv| = v = sqrt(v_(x)^(2) + v_(y)^(2)+v_(z)^(2))`
`(d|vecv|)/(dt) = 1/(2sqrt(v_(x)^(2) + v_(y)^(2)+v_(z)^(2)))`
` xx [2v_x (dv_x)/(st) + 2v_y (dv_y)/(st) + 2v_z (dv_z)/(st)] = (vecv*veca)/v`
which is a variable quantity.
`|(dvecv)/(dt)| = |veca|,` which would be constant.
`(dv^2)/(dt) = (d(v_(x)^(2) + v_(y)^(2)+v_(z)^(2)))/(dt) = 2vecv*veca` (a variable quantity)
`(d(vecv//v))/(dt) = d/(dt)[(v_xhati+v_yhatj+v_zhatk)/(sqrt(v_(x)^(2) + v_(y)^(2)+v_(z)^(2)))]` (a variable quantity).