KAHIBARO
Table of Contents
Vector nature of acceleration
In two and three dimensional motion, acceleration tells us how the velocity vector changes with time. Since velocity has both magnitude and direction, acceleration can appear when speed changes, when direction changes, or when both change at the same time.
If the position of a particle is $\vec{r}(t)$ and its velocity is $\vec{v}(t)$, then the acceleration vector is defined by
$$
\vec{a}(t) = \frac{d\vec{v}}{dt} = \frac{d^2 \vec{r}}{dt^2}
$$
This means acceleration is the time rate of change of velocity, not simply the rate of change of speed.
Important idea: an object can have acceleration even if its speed stays constant, as long as its direction changes.
Average and instantaneous acceleration
Over a finite time interval $\Delta t$, the average acceleration is
$$
\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}
= \frac{\vec{v}_2 - \vec{v}_1}{t_2 - t_1}
$$
This vector points in the direction of the change in velocity, not necessarily in the same direction as the motion itself.
The instantaneous acceleration is obtained by taking the limit as the time interval becomes very small:
$$
\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t}
$$
This gives the acceleration at one specific instant.
Components of acceleration
In Cartesian coordinates, if
$$
\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j} + z(t)\hat{k}
$$
then the velocity is
$$
\vec{v}(t) = \frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}
$$
and the acceleration is
$$
\vec{a}(t) = \frac{d^2x}{dt^2}\hat{i} + \frac{d^2y}{dt^2}\hat{j} + \frac{d^2z}{dt^2}\hat{k}
$$
So we usually write
$$
\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}
$$
with
$$
a_x = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}, \quad
a_y = \frac{dv_y}{dt} = \frac{d^2y}{dt^2}, \quad
a_z = \frac{dv_z}{dt} = \frac{d^2z}{dt^2}
$$
Each component describes how the corresponding velocity component changes with time.
In component form, acceleration is found by differentiating each velocity component separately.
Magnitude of the acceleration vector
The size, or magnitude, of the acceleration vector is
$$
|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}
$$
In two dimensions this becomes
$$
|\vec{a}| = \sqrt{a_x^2 + a_y^2}
$$
This gives the total amount of acceleration, combining all directions.
Interpreting the direction of acceleration
The direction of the acceleration vector shows how the velocity vector is changing. This is very important in two and three dimensional motion.
If acceleration points in the same direction as velocity, the object speeds up.
If acceleration points opposite to velocity, the object slows down.
If acceleration is perpendicular to velocity, the speed may remain constant while the direction changes.
In general, acceleration can have one part parallel to the motion and another part perpendicular to the motion. Then both speed and direction change at once.
Example in component form
Suppose a particle has velocity
$$
\vec{v}(t) = (3t)\hat{i} + (4 - 2t)\hat{j}
$$
Then the acceleration is found by differentiating each component:
$$
\vec{a}(t) = \frac{d}{dt}\big[(3t)\hat{i} + (4 - 2t)\hat{j}\big]
= 3\hat{i} - 2\hat{j}
$$
So the acceleration is constant.
Its magnitude is
$$
|\vec{a}| = \sqrt{3^2 + (-2)^2} = \sqrt{13}
$$
This example shows that even when the velocity changes with time, the acceleration can still be constant.
Relation to motion in a plane
In motion on a plane, the velocity might point one way while acceleration points another way. For example, if an object moves to the right and upward, its acceleration could point downward, sideways, or in any other direction depending on how the velocity is changing.
The important point is that acceleration is determined by changes in velocity, not by the current position alone.
This drawing shows that the acceleration vector does not need to point along the velocity vector.
Constant acceleration in two and three dimensions
If acceleration is constant, then each component is constant:
$$
\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}
$$
In that case, the velocity components change linearly with time:
$$
v_x = v_{x0} + a_x t, \quad
v_y = v_{y0} + a_y t, \quad
v_z = v_{z0} + a_z t
$$
and the position components follow
$$
x = x_0 + v_{x0} t + \frac{1}{2} a_x t^2
$$
$$
y = y_0 + v_{y0} t + \frac{1}{2} a_y t^2
$$
$$
z = z_0 + v_{z0} t + \frac{1}{2} a_z t^2
$$
These equations are just the component version of constant acceleration motion.
For constant acceleration, apply the kinematic equations separately to each coordinate direction.
Common special cases
Many motions can be understood by looking at acceleration components.
| Situation | Acceleration vector |
|---|---|
| Motion with no change in velocity | $\vec{a} = \vec{0}$ |
| Straight line speeding up | $\vec{a}$ parallel to $\vec{v}$ |
| Straight line slowing down | $\vec{a}$ opposite to $\vec{v}$ |
| Turning motion at constant speed | $\vec{a}$ perpendicular to $\vec{v}$ |
These cases help build physical intuition about what acceleration means.
Units of acceleration
The SI unit of acceleration is meters per second squared:
$$
\mathrm{m/s^2}
$$
This unit means the velocity changes by a certain number of meters per second during each second.
Acceleration has units of velocity divided by time, so its SI unit is $\mathrm{m/s^2}$.
Final perspective
Acceleration vectors are essential in multidimensional motion because they describe exactly how velocity changes in both size and direction. In two and three dimensions, the clearest method is usually to work with components. Once the acceleration components are known, the motion can be analyzed coordinate by coordinate, and then combined into the full vector description.