Question

The equation Bold r (t )equals(8 t plus 9 )Bold i plus (2 t squared minus 8 )Bold j plus (6 t )Bold k is the position of a particle in space at time t. Find the​ particle's velocity and acceleration vectors. Then write the​ particle's velocity at t equals 0 as a product of its speed and direction.

Answer 1

Step-by-step explanation:

It is given that, the position of a particle as as function of time t is given by :

`r(t)=(8t+9)i+(2t^2-8)j+6tk`

Let v is the velocity of the particle. Velocity of an object is given by :

`v=(dr(t))/(dt)`

`v=(d[(8t+9)i+(2t^2-8)j+6tk])/(dt)`

`v=(8i+4tj+6k)\ m/s`

So, the above equation is the velocity vector.

Let a is the acceleration of the particle. Acceleration of an object is given by :

`a=(dv(t))/(dt)`

`a=(d[8i+4tj+6k])/(dt)`

`a=(4j)\ m/s^2`

At t = 0,
`v=(8i+0+6k)\ m/s`

`v(t)=√(8^2+6^2) =10\ m/s`

Hence, this is the required solution.