Question

How would you find the rate of acceleration?

Answer 1

Hint: Try and recall the kinematic equations of motion that relate the fundamental physical quantities like distance, speed and time to acceleration. Additionally, recall Newton’s $2^{nd}$ law of motion for massive systems that brings into picture an interactive force in terms of mass and acceleration.

Formula Used:
$v=u+at$
$s=ut+\dfrac{1}{2}at^2$
$v^2=u^2 +2as$
$F=ma$

Complete Solution:
Let us begin by understanding what acceleration means following which we can obtain expressions that can help us deduce conditional estimates for the same. We will then look at 3 kinematic equations and 1 force related equation.

Acceleration is a physical quantity that is associated with the motion of an object. It is a vector quantity as it possesses both magnitude and direction. Mechanically, acceleration is defined as the rate at which the velocity of a moving object changes over time., i.e.,
$a = \dfrac{\Delta v}{t} = \dfrac{v-u}{t} \Rightarrow v = u+at$, which is the first kinematic equation of motion.
Here, v is the final velocity, u is the initial velocity and t is the time through which the object accelerates.

The next kinematic equation is given as
$s=ut+\dfrac{1}{2}at^2$, which takes into account the distance s that the object travels under the influence of acceleration.

The final kinematic equation is given as
$v^2=u^2 +2as$, which is a time independent equation.

The final equation that takes into consideration the fact that the orientation of an object’s acceleration is given by the orientation of the net force acting on it. For an object with a mass m, if the net force acting on it is F, then the acceleration which acts on the object is given as the force acting on a unit mass of the object, i.e.,
$a = \dfrac{F}{m}$

This is a consequence of Newton’s second law of motion which states that the force acting on an object is equivalent to the mass of that object times its acceleration.
$F=ma$

Based on available data and given (or missing) quantities, one can appropriately choose which equation to use to deduce the acceleration of the object with the data at hand.

Note:
Kinematic equations describe the motion of objects without considering the contribution of forces that cause their state of rest or motion. If the force responsible for the state of rest or motion of an object is not available, then kinematic equations are a great way to evaluate the objects’ motion.
It is also important to remember that the kinematic equations of motion can be used for not only linear acceleration but also for gravitational acceleration and retardation calculations. In such a case, it assumes ‘g’ and ‘-a’ values respectively.