Find the velocity, acceleration, and speed of a particle with position function

Question

Find the velocity, acceleration, and speed of a particle with position function r(t) = <-7tsin(t), -7tcos(t), 4t^2>

 

v(t) = <?, ?, ?>

a(t) = <?, ?, ?>

|v(t)|= ?

Answer 1

Hi Ally

 

V = derivative of r wrt time

 

V = dr/dt

V= d/dt(-7tsint) = -7t. cos t -7sint

and acceleration  is derivative of velocity  wrt time

 

dV/dt =d/dt ( -7t.cost -7sint) or second derivative of position 

 

= 7t sint -7cost - 7 cost

= 7t sint -14 cost 

Similarly for other two functions

Answer 2

Ally, 

 

When a vector is given in terms of the CONSTANT unit vectors i, j, and k, as in this problem, all you have to do is differentiate each component.

 

r = <x(t), y(t), z(t)>

 

v = <dx/dt, dy/dt, dz/dt>

 

a = <d²x/dt², d²y/dt², d²z/dt²>

 

v = |v| = √[(dx/dt)² + (dy/dt)² + (dz/dt)²]