Motion in 2 and 3 dimensions
Acceleration
Acceleration is defined as the derivative of the vector-quantity velocity, $$\vec a=\frac{\dd \vec v}{\dd t}$$ and thus is a vector in its own right, describing the rate of change of the velocity.
By explicit differentiation of the components we can get quite a simple expression: $$\begin{align*} \vec{a}&=a_x \uvi +a_y \uvj +a_z \uvec k\\ &= \frac{\dd v_x}{\dd t} \uvi +\frac{\dd v_y}{\dd t} \uvj +\frac{\dd v_z}{\dd t} \uvec k\\ &= \frac{\dd ^2x}{\dd t^2} \uvi +\frac{\dd ^2y}{\dd t^2} \uvj +\frac{\dd ^2z}{\dd t^2} \uvec k\,. \end{align*}$$
Parallel and Perpendicular Components of Acceleration
The vector $\vec a$ does not necessarily have the same direction as the vector $\vec v$, which means that the interpretation is more complicated then for the one-dimensioanl case. One way to understand this better, is by looking at the components of acceleration which are parallel and perpendicular to the direction of the velocity. This actually gives us an understanding how the velocity is changing.
For a particle travelling along a curved path $a_\parallel$ is in the same direction as the instantaneous velocity and is tangent to the actual path. The other conponent $a_\perp$ is perpendicular to the instantaneous velocity and thus also perpendicular to the direction of the actual path.
Consider two extreme cases as shown in the diagram above.
- a) $\vec a$ is parallel to both $\vec v_1$ and $\vec v_2$, i.e., its perpendicular component is zero. In this case, the particle’s speed is increasing, but its direction is not changing.
- b) $\vec a$ is perpendicular to the path, and thus very nearly perpendicular to $\vec v_1$ and $\vec v_2$. Looking at the rate of change of the instantaneous speed squared, using the product rule for differentiation of vectors we see that $$\frac{\dd v^2}{\dd t}=\frac{\dd \vec v}{\dd t}\cdot\vec v+\vec v \cdot\frac{\dd \vec v}{\dd t} =2\vec a \cdot \vec v\,,$$ and thus the speed is a constant.
It follows that
- when $\vec a$ is parallel to $\vec v$ the effect is to change the speed but not the direction of the particle;
- when $\vec a$ is perpendicular to $\vec v$, the effect is to maintain constant speed but change the direction in which the particle is moving – the particle will move along a curved path with constant speed.
In general, $\vec a$ may have both parallel and perpendicular components, in which case, the above statements apply to the individual components.
What is happening to the speed?
To test your understanding of the material in this section, look at the following three cases. Can you say how the speed (magnitude of velocity) varies with time?
A detailed analysis of a more complex example
Once you have listened to the talklet and understood it,
click here for an interactive applet to illustrate
this further.