Acceleration

Basic Electrical Engineering > Fundamental of Physics > Motion

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When the velocity of a particle changes with time, the particle is said to be accelerating.

For example, the velocity of a car increases when you step on the gas and decreases when you apply the brakes. It is easy to quantify changes in velocity as a function of time in exactly the same way we quantify changes in position as a function of time.

Definition of Acceleration

The average acceleration of the particle is defined as the change in velocity $\Delta v_x$ divided by the time interval $\Delta t$ during which that change occurred:

$$ a = {{\Delta v_x} \over {\Delta t}}$$

Unit of Acceleration

The SI unit of acceleration is meters per second squared $(m/s^{2})$.

In the following example, we will see how the velocity of a particle changes while the particle is moving. Suppose a particle moving along the x axis has a velocity $v_{xi}$ at time $t_i$ and a velocity $v_{xf}$ at time $t_f$. For motion along an axis, the average acceleration $a_{avg}$ over a time interval $\Delta t$ is

$$ \bbox[10px,border:1px solid grey]{a_{avg} = { v_{xf} - v_{xi} \over t_f - t_i} = { \Delta vx \over \Delta t}}$$

where the particle has velocity $v_{xi}$ at time $t_f$ and then velocity $v_{xf}$ at time $t_i$. The instantaneous acceleration (or simply acceleration) is

$$ v = { dv \over dt}$$

In words, the acceleration of a particle at any instant is the rate at which its velocity is changing at that instant. Graphically, the acceleration at any point is the slope of the curve of v(t) at that point.

$$ v = { dv \over dt} = {d ({dx \over dt}) \over dt} = {d^2x \over d^2t}$$

In words, the acceleration of a particle at any instant is the second derivative of its position x(t) with respect to time.

Acceleration has both magnitude and direction (it is yet another vector quantity). Its algebraic sign represents its direction on an axis just as for displacement and velocity; that is, acceleration with a positive value is in the positive direction of an axis, and acceleration with a negative value is in the negative direction.

It might be easier to interpret these units if you think of them as meters per second per second. For example, suppose an object has an acceleration of $2 m/s^2$. You should form a mental image of the object having a velocity that is along a straight line and is increasing by 2 m/s during every 1-s interval. If the object starts from rest, you should be able to picture it moving at a velocity of $+2 m/s$ after $1 s$, at $+4 m/s$ after $2 s$, and so on. In some situations, the value of the average acceleration may be different over different time intervals. It is therefore useful to define the instantaneous acceleration as the limit of the average acceleration as $\Delta t$ approaches zero.

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