We know that acceleration is a vector quantity; hence it can be positive or negative, but what about the average acceleration, can average acceleration be negative?

**The average of acceleration is a total change in velocity within a given time; the average is also a vector quantity. From this, we can say that it can be negative or positive. Let’s discuss acceleration, an average of acceleration, instantaneous acceleration, and the conditions for an average of acceleration to be negative**.

**Acceleration**

In simple words, acceleration is how the velocity of an object changes during its motion in a particular interval of time. The SI unit of acceleration is m/s^{2}. Acceleration is a vector quantity so that it can be negative and positive. Mathematically it is denoted as,

Where v – velocity

t – Time

Now you know that the acceleration can be negative or positive. However, the question will arise in your mind, What are those conditions in which acceleration can be positive or negative, and what is the impact of positive and negative acceleration on an average acceleration? We will discuss these questions in the course of the article.

**Average acceleration **

**Average acceleration is the total change in velocity that occurred during the motion and the time required to complete that motion.** Consider a man walking on a street; he starts to move with the initial rate ‘u’ and after some time ‘t’ his final velocity is ‘v’. During the motion, he changes his velocity several times in between initial and final velocity. So to find out the total change in velocity during the motion, we subtract his initial velocity from the final velocity, i.e.,

Total change in velocity = ∆v = v – u

We know that acceleration is a change in velocity with respect to time. So if we divide the total change in speed of man by the time taken to complete motion, then we get the average acceleration of the man. [latex]\Delta t[/latex]

u – initial velocity

[latex]\Delta t[/latex] – average time

**Instantaneous Acceleration**

Instantaneous acceleration is the acceleration at a particular instant of time. Consider a man performing motion on the street, v1, v2, v3…………. is the velocity of that man at time t1, t2, t3……… So if we want to find out the acceleration at instant t3, then by taking the ratio of velocity and acceleration at that particular instant, we can find out the acceleration at t3. Generalized formulae to calculate the instantaneous acceleration is

where v – final velocity

where, dv – instantaneous velocity

dt – particular instant of time

**Conditions for negative average acceleration **

Let’s understand first what negative and positive signs indicate. In physics, when we deal with vector quantities, it is essential to know about the directions of those quantities, so negative and positive signs show the direction of the physical quantities in the physical world.

Commonly north, south, east, west, right, left, up, down are all the notions of direction in the physical scenario. In mathematics, we often see a plane is divided into four quadrants. –ve sign shows the left (-x axis) or downside (-y-axis) of a plane or +ve sign shows the right or upside of a plane; we borrow that concept in the physical world, and say –ve quantity means left, or downside and +ve quantity means right or upside.

As we discussed earlier, acceleration is a vector quantity, and it can be negative and positive in direction. And we also know that average acceleration is the total change in velocity in a given interval of time.

lets discuss about sign of average acceleration So average acceleration is negative in the following two conditions,

**When the magnitude of velocity is negative –**

The magnitude of velocity is negative when the object is moving in a positive direction and slowing down. In this condition object is decelerating, and the direction of deceleration is opposite to the direction of velocity

**When the direction of velocity is negative –**

When the object performs motion in the negative direction and speeds up, the velocity direction is negative, and the acceleration is negative. Consider a car is moving in negative x-direction and speeding up in that case, acceleration is in the direction of velocity, and both have negative direction. Hence the average acceleration of the car is negative as the vehicle moves in a negative direction.

- We can break down the above figure in four cases,
**case1**:- In this case, the car is moving in the positive x-direction and speeding up, so the velocity and acceleration both are in the same direction, and both are positive.**Case2**:- In the second case car is moving in a negative direction and speeding up, so the acceleration is along the path of velocity and negative.**Case3**:- In the third car’s case, the vehicle is moving in a positive direction but slowing down. Here, the direction of velocity and acceleration is the opposite, as you can see in fig. The sign of acceleration is negative.**Case4**:- In the fourth case, the car moves negatively and slows down. Here the direction of velocity is negative, and the direction of acceleration is positive, so we can say that the car has positive acceleration.