4.2 Acceleration vector  (Page 5/4)

Instantaneous acceleration

In addition to obtaining the displacement and velocity vectors of an object in motion, we often want to know its acceleration vector    at any point in time along its trajectory. This acceleration vector is the instantaneous acceleration and it can be obtained from the derivative with respect to time of the velocity function, as we have seen in a previous chapter. The only difference in two or three dimensions is that these are now vector quantities. Taking the derivative with respect to time $\overrightarrow{v}(t),$ we find

The acceleration in terms of components is

Also, since the velocity is the derivative of the position function, we can write the acceleration in terms of the second derivative of the position function:

Finding a particle acceleration

Strategy

Solution

The velocity function is linear in time in the x direction and is constant in the y and z directions.

(b) Taking the derivative of the velocity function, we find

The acceleration vector is a constant in the negative x -direction.

(c) The trajectory of the particle can be seen in [link] . Let’s look in the y and z directions first. The particle’s position increases steadily as a function of time with a constant velocity in these directions. In the x direction, however, the particle follows a path in positive x until t = 5 s, when it reverses direction. We know this from looking at the velocity function, which becomes zero at this time and negative thereafter. We also know this because the acceleration is negative and constant—meaning, the particle is decelerating, or accelerating in the negative direction. The particle’s position reaches 25 m, where it then reverses direction and begins to accelerate in the negative x direction. The position reaches zero at t = 10 s.