On the basis of the time interval considered for calculation, the acceleration is classified as average acceleration and instantaneous acceleration. In this article, we are discussing instantaneous acceleration in detail.
Contents:
What is Instantaneous acceleration?
Instantaneous acceleration is the acceleration of the object at a specific instant during the motion. It indicates the change in velocity per unit time measured for a very small interval ‘dt’.
The object moving in a straight line may undergo an increase or decrease in acceleration or it may move with a uniform acceleration or zero acceleration. Thus in such cases, the average acceleration does not describe the motion of the object at every instant.
The average acceleration only provides the mean value of the acceleration instead of the actual acceleration of the object during the motion. While the instantaneous acceleration gives the exact acceleration at every instant during the motion.
In the above accelerationtime graph, the `a_{\text{average}}` indicates the average acceleration of the object throughout the motion while the `a_{1}` and `a_{2}` indicates the instantaneous acceleration of the object at specific instant `t_{1}` and `t_{2}` respectively.
Equations:
In the form of limits, the instantaneous acceleration of the object is given by,
`a_{\text{instantaneous}}` = `\lim_{\Delta t \to 0} \frac{V_{(t + \Delta t)}V_{t}}{\Delta t}` 
In the form of differentiation, the instantaneous acceleration of the object is given by,
`a_{\text{instantaneous}}` = `\frac{dV}{dt}` 
How to find the instantaneous acceleration?
It can be found by use of the following two methods:
1] Analytical method:
This method is used to find the instantaneous acceleration when the equation of velocity in terms of time is given. It can be solved by using the method of limits or by using differentiation.
By using limits, the instantaneous acceleration is calculated as,
`a_{\text{instantaneous}}` = `\lim_{\Delta t \to 0} \frac{V_{(t + \Delta t)}V_{t}}{\Delta t}`
By using differentiation, the instantaneous velocity can be calculated as,
`a_{\text{instantaneous}}` = `\frac{dV}{dt}`
2] Graphical method:
This method is used for calculating instantaneous acceleration from the velocitytime graph.
How to find instantaneous acceleration from a velocitytime graph?
The instantaneous acceleration from the different profiles of a velocitytime graph is calculated as follows:
Case 1] When a velocitytime graph is linear:
If the velocitytime plot has a linear nature, then it means that the object is moving with constant acceleration throughout its motion.
Thus in this case, the instantaneous acceleration is equal to the average acceleration of the object.
`V_{\text{Instantaneous}}` = `V_{\text{Average}}`
`V_{\text{Instantaneous}}` = `\frac{\Delta V}{\Delta t}`
`V_{\text{Instantaneous}}` = `\frac{V_{2}V_{1}}{t_{2}t_{1}}`
Case 2] When velocitytime profile is nonlinear:
If the velocity time profile is nonlinear thus it means that the acceleration of the objeat is changing throughout the motion.
Thus in this case, the instantaneous acceleration of an object is different from the average acceleration. From such a graph, the instantaneous acceleration can be calculated by the use of the following steps:
Step 1] Locate a point of a curve at a required time:
Step 2] At this point draw a tangent to the curve:
Step 3] Find the slope of the tangent: The instantaneous acceleration is the slope of the tangent drawn to the velocitytime curve at that instant.
∴ The instantaneous acceleration is given by,
`a_{\text{Instantaneous}}` = Slope(12) = `\frac{V_{2}V_{1}}{t_{2}t_{1}}`
Average acceleration vs Instantaneous acceleration:
Sr. No.  Average acceleration  Instantaneous acceleration 

1  Average acceleration gives the change in velocity within a considerable duration.  It gives the acceleration of the object at a particular instant. 
2  It is calculated for a considerable time interval ‘`\Deltat`’.  It is calculated for the smallest interval of time ‘dt’. 
3  It gives the average value of acceleration during a specified interval.  It gives exact acceleration at any instant. 
4  It is given by, `a_{\text{average}}` = `\frac{\DeltaV}{\Delta t}`  It is given by, `a_{\text{instantaneous}}` = `\frac{dV}{dt}` 
Solved examples:
1] The particle is moving in a straight line with the velocity of `(5t^{2} +3t)` m/s. What is the instantaneous acceleration of the particle at t = 45.0s? 
Given:
V = (`5t^{2}` +3t) m/s
t = 45 seconds
Solution:
Method 1: Using limits
The instantaneous acceleration of the object is given by,
`a_{t}` = `\lim_{\Delta t \to 0} \frac{V_{(t + \Delta t)}V_{t}}{\Delta t}`
`a_{t}` = `\lim_{\Delta t \to 0} \frac{[5(t + \Delta t)^{2} + 3(t + \Delta t)] – [5t^{2} +3t]}{\Delta t}`
`a_{t}` = `\lim_{\Delta t \to 0} \frac{5t^{2} + 10t\Delta t + 5\Delta t^{2}+3t+3\Delta t5t^{2}3t}{\Delta t}`
`a_{t}` = `\lim_{\Delta t \to 0} \frac{5\Delta t^{2}+10t\Delta t+3\Delta t}{\Delta t}`
`a_{t}` = `\lim_{\Delta t \to 0} 5\Delta t` + 10t + 3
`a_{t}` = 10t + 3
Now the acceleration of the object at t = 45 seconds is given by,
`a_{(t=45)}` = 10(45) + 3
`a_{(t=45)}` = 453 m/s² 
Method 2: Using differentiation
`a_{t}` = `\frac{dV}{dt}`
`a_{t}` = `\frac{d}{dt}(5t^{2} + 3t)`
`a_{t}` = 10t + 3
The acceleration at t = 45 seconds is given by,
`a_{(t=45)}` = 10(45) + 3
`a_{(t=45)}` = 453 m/s² 
2] From the given velocitytime plot, find the acceleration of the object at t = 3.5 seconds. 
Given:
t = 3.5 seconds
Solution:
Step 1] Locate point on curve at time t = 3.5 seconds:
Step 2] Draw a tangent to the curve at point x:
Step 3] Find the slope of the tangent:
∴ The instantaneous acceleration at t = 3.5 second is given by,
`a_{(t=3.5)}` = Slope(12) = `\frac{V_{2}V_{1}}{t_{2}t_{1}}`
From above figure,
`V_{2}` = 0.5 m/s, `V_{1}`= 3 m/s,
`t_{2}` = 6 seconds, `t_{1}` = 1 second.
`a_{(t=3.5)}` = `\frac{0.53}{61}`
`a_{(t=3.5)}` = – 0.5 m/s²
Thus at t = 3.5 seconds, the object has a retardation of 0.5 m/s². 
3] A particle starting from rest is moving at the velocity of (6t² + 2t) m/s. If the average acceleration of the particle is 6 m/s², then find the time when average acceleration is equal to instantaneous acceleration. 
Given:
V = (`6t^{2}` + 2t) m/s
`a_{\text{average}}` = 6 m/s²
Solution:
The instantaneous acceleration of the particle is given by,
`a_{\text{instantaneous}}` = `\frac{dV}{dt}`
`a_{\text{instantaneous}}` = `\frac{d}{dt}(6t^{2} + 2t)`
`a_{\text{instantaneous}}` = 12t + 2
The time at which the `V_{\text{average}}` is equals to the `V_{\text{instantaneous}}` is given by,
`a_{\text{average}}` = `a_{\text{instantaneous}}`
6 = 12t + 2
t = 0.34 seconds 
FAQs:

Why instantaneous acceleration is important?
It gives the acceleration of the object at a specific instant during the motion.

Is instantaneous acceleration always changing?
For the object moving with constant acceleration or constant retardation, the instantaneous acceleration never changes while for the object moving with continuously changing its acceleration, the instantaneous acceleration of the object always changes.
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