What is Instantaneous acceleration, how is it calculated and how can a moving object have Instantaneous acceleration at specific time?

In order to understand the concept of acceleration, be it average of instantaneous, one must have a fundamentally strong understanding of basics of Kinematics, such as the concept of rest, motion, speed, velocity etc. Before discussing translator motion, it is necessary to first grasp the meaning of the term “Motion” from a physics standpoint. The question arises “When can we claim that a body is in motion or at rest?”.In non-professionals terms, one might say that if a body does not change its position with respect to time (or with passage of time), hence the body is at rest. We say the body is in motion if it is moving about and changing positions with regard to time.

Consider a book that has been put on top of a table. If one were to sit close to the table, the book would appear to be at rest because it was not moving. However, if the same individual observed the same table while sitting on the moon, the conclusion would be different since the Earth moves in relation to the moon (it changes positions), the room moves, and the book moves as well.Therefore, it can be concluded that the book is in motion. Thus, it needs to be understood that the state of motion or rest is with respect to a reference frame and an observer.

Therefore, Motion is a combined property of an object of our interest and an observer. Thus, the terms “in motion” or “at rest” are meaningless without the presence of an observer. Strictly speaking, nothing is ever at absolute rest or in absolute motion.

Let us now discuss the concept of distance and displacement. The fundamental unit of distance or displacement is length and always measured in meters in SI unit. To put it simply, distance is a scalar quantity whereas displacement is a vector quantity. Let us assume that a person walked 40 meters due East and then turned right to walk 30 meters due South, and stopped there. It can be understood that from the point of origin, the person has walked 70 meters. Thus, the distance travelled by the person is 70 m. However, it is not the same as displacement, as it would be calculated as the shortest distance from the point of origin, traversed by the person. In this case, it is 50 m. The answer is still not complete, as displacement is a vector term. Thus, we have to conclude the statement by saying the displacement of the person is 50 m due South East.

From the concept of distance and displacement, comes the concept of speed and velocity. Speed is a scalar term while velocity is a vector quantity. Speed is derived from distance travelled whereas velocity is a displacement dependent quantity. For the same example given above, let us consider that the person travels 40 m due East in 4 seconds while he travels 30 meters due south in 3 seconds.

For the computation of speed, it can be calculated to be the ratio of distance over time taken. Here in the case, it is 10 meters per second. However, displacement over time gives the velocity of the body. Therefore, the velocity of the person is the ratio of 50/7 = 7.142 meters per second, due South East.

Now here, an important concept of average and instantaneous must be introduced. Let us try and define average and instantaneous velocity, and with that concept we will built up the understanding of instantaneous acceleration as well. Average velocity is the velocity of a body in a time interval, whereas the instantaneous velocity is the velocity of the body at any given instant of time.

Acceleration is a velocity dependent quantity which tries to explain how the behavior of velocity was in either a given time interval or at any given instant of motion of the body. If the velocity of a body remains constant as with the passage of time, it is described to be moving with a constant velocity. However, if the velocity of the body changes with time, then it is said to be accelerating. Thus, acceleration is the rate of change of velocity of a body. As evident, it’s a velocity dependent parameter and thus is also a vector. If a body accelerates from v1 to v2, in a time interval from t1 to t2, thus the ratio of change in velocity over the time interval is the average acceleration. However, Instantaneous acceleration a(t) is actually a continuous function of time. It gives the acceleration at any specific time during the motion of the body. It is computed from the derivative of the velocity function. Simply understood, instantaneous acceleration is the slope of the velocity vs time graph at any given instant of time.