2.4 Average and instantaneous acceleration

  Motion in One Dimension

Acceleration: is defined as the rate of change of velocity.

In other words, acceleration is defined as the ratio of change in velocity to the time interval in which this change occurs.

Mathematically, $${\overrightarrow a _{avg}} = \frac{{\Delta \overrightarrow v }}{{\Delta t}}$$

Instantaneous acceleration: is defined as the acceleration of a particle at a particular instant of time.
$$\overrightarrow a = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta \overrightarrow v }}{{\Delta t}}$$$$\overrightarrow a = \frac{{d\overrightarrow v }}{{dt}}$$

Note:

• Acceleration is a vector quantity
• Different terminologies of acceleration,
• Acceleration: when acceleration and velocity is in same direction
• Retardation: when acceleration and velocity is in opposite direction
• SI unit of acceleration is $m/{s^2}$
• Acceleration can be positive, negative or zero

Question: The position of a particle moving along $x-$ axis at any time $t$ is given by,
$$x = 3{t^2} + 5t - 6$$
Find,
(a). Instantaneous velocity at $t=5s$
(b). Average velocity between $t=3$ to $5s$

Solution: The position of a particle is given as,
$$x = 3{t^2} + 5t - 6$$
(a). For instantaneous velocity we will differentiate the position of a particle wrt time $t$.
$${v_{ins}} = \frac{{dx}}{{dt}} = 6t + 5$$
At $t=5\ s$, $${v_{ins}} = 6(5) + 5$$$${v_{ins}} = 35\,m/s$$
(b). Average velocity is given as,
$${v_{avg}} = \frac{{{\text{Displacement}}}}{{{\text{Total time}}}} = \frac{{\Delta s}}{{\Delta t}}$$
Displacement is given by,
$$x = {x_f} - {x_i}$$$$x = \left[ {3{{\left( 5 \right)}^2} + 5\left( 5 \right) - 6} \right] - \left[ {3{{\left( 3 \right)}^2} + 5\left( 3 \right) - 6} \right]$$$$x = \left[ {75 + 25 - 6} \right] - \left[ {27 + 15 - 6} \right]$$$$x = 94 - 36$$$$x = 58\,m$$
Time interval is given by, $$\Delta t = {t_f} - {t_i}$$$$\Delta t = 5 - 3$$$$\Delta t = 2s$$
So, average velocity is given by,
$${v_{avg}} = \frac{{\Delta s}}{{\Delta t}}$$$${v_{avg}} = \frac{{58}}{2}$$$${v_{avg}} = 29\,m/s$$