2D velocity and acceleration vector from position vector + dot product of acceleration and velocity.

2D velocity and acceleration vector from position vector + dot product of acceleration and velocity.

Given the position vector as a function of time for a particle, we compute the velocity vector and we compute the acceleration vector as time derivatives of the position vector.

Next, we compute the velocity and acceleration vectors at a given time t=1s.

Finally, we are asked for the dot product of velocity and acceleration, and this gives us a way to tell whether or not the particle is speeding up or slowing down at the moment of interest. If the dot product of velocity and acceleration is positive, this means the parallel component of the acceleration vector points in the same direction as the velocity vector, so the particle is speeding up. If the dot product of acceleration and velocity is negative, this means the parallel component of the acceleration vector points opposite the velocity vector, and the particle is slowing down.