Can Acceleration Be Negative? To answer this question we have to understand, what is acceleration?

**We know that, acceleration is a vector quantity, and a vector quantity can be positive or negative. Therefore we can say that acceleration of an object can be negative as well as positive.**

However, now the question arises, in what conditions acceleration of an object is negative and when is it positive? Here we are going to discuss those conditions in which acceleration may be negative or positive.

**Acceleration: A vector quantity**

Acceleration is a vector quantity; therefore, it has magnitude as well as direction. Consider a biker riding a bike or a person driving a car, then that bike or car moves with a continuously changing velocity. The rate of change of velocity, that car or bike, per unit time is called acceleration. When an object changes velocity, it starts to accelerate, maybe in the direction of velocity or opposite to velocity. Mathematically it is expressed as,

[latex]A_{ave}=\frac{\vec{v_{f}}-\vec{v_{i}}}{\Delta t}[/latex]

[latex]A_{ave}[/latex] – average acceleration

[latex]\vec{v}_{f}[/latex] – final velocity

[latex]\vec{v}_{f}[/latex] – initial velocity

This equation gives the magnitude of acceleration.

**Direction of Acceleration**

The object always accelerates in the direction of the net force applying to it. Whether the acceleration of an object is positive or negative is depends on the direction of acceleration, and the direction depends on the following two factors:

- speeding up or slowing down of an object
- +ve or -ve direction of motion [here we consider, left to right as +ve direction, and right to left -ve direction. Similarly, up is +ve, and down is – ve ]

Consider a car is moving on a road, to determine the direction of acceleration of the car, above two factors make four combinations to describe the motion of a car.

**The car is moving in +ve direction and speeding up**

In this case, the velocity and acceleration of a car are in the same direction. Force acting on a car is in the +ve direction, and the acceleration of a car is positive.

**The car is moving in +ve direction and slowing down.**

In this case, the car’s velocity is in the +ve direction, and acceleration is in the –ve direction. The friction force is responsible for slowing down a car, and it is opposite to the direction of velocity. In this case, the acceleration is negative

**The car is moving in –ve direction and speeding up**

The car is moving from right to left direction, i.e., in –ve direction. The car is speeding up in the negative direction, so the acceleration is also negative. Acceleration is negative in this case.

**The car is moving in the –ve direction and slowing down.**

Here velocity of the car is in the –ve direction. The car is slowing down, which means some force acting in the opposite direction on a car. Therefore the acceleration of a car is in a +ve direction.

**From the above discussion, we conclude that acceleration is negative in two cases**

**when an object is moving in +ve direction and slowing down **

**when an object is moving in –ve direction and speeding up**

## Can acceleration be negative when velocity is zero?

We know that acceleration is a rate of change of velocity with time. This change may be in the form of speeding up or slowing down.

When the acceleration is negative it means the object is speeding up in the -ve direction or slowing down in the +ve direction. when object is slowing down force is opposing its motion, and the direction of acceleration is along the direction of force i.e. in -ve direction. After some time the body comes to rest and its velocity becomes zero, but at that point it still has negative acceleration.

To better understand this concept remember the motion of pendulum or the motion of ball in vertical direction, at certain height the ball comes to rest and its velocity becomes zero. But it still has acceleration in the downward direction, similarly, in pendulum, at its extreme position velocity becomes zero but it still has acceleration in the direction of restoring force. Hence, this proves that acceleration can be negative when velocity becomes zero.

**Some examples having negative acceleration**

**Applying breaks to a moving car**

Consider a car is moving in a +ve x-direction with continuously changing velocity and acceleration. After applying breaks, friction force builds against the direction of velocity, and velocity starts to decreases. As the velocity starts to decrease, the direction of acceleration gets changes from +ve x-direction to –ve x-direction; this happens because acceleration is always in the direction of the force. Hence a moving car has a negative acceleration when we apply breaks to stop motion.

**The motion of a stretched spring **

stretched spring has a restoring force opposite to the direction of motion. When a stretched spring is released, it performs SHM. Restoring forces always oppose the motion of spring and decrease the velocity.

Spring has both +ve as well as –ve type of acceleration. When spring gets stretched from its mean position, it moves in a +ve x-direction but slows down due to restoring force. In that case, its acceleration is opposite to the direction of motion, i.e., in –ve x-direction. Similarly, when spring gets compressed, its acceleration is +ve x-direction and motion in –ve x-direction from the mean position.

**Throwing a ball in an upward direction**

When we throw a ball in an upward direction, it moves opposite to gravitational force. After achieving some height, it starts to fall towards earth in the direction of gravitational force. In the first half of motion, gravitational force continuously opposes the motion of a ball, and its acceleration is in the downward direction. Moreover, according to the above analysis, **when an object moves in +ve direction (here, in the upward direction) and slows down, its acceleration is negative**.

In the second half of a motion, the ball’s velocity gets zero after attaining a certain height, and it starts to fall towards the earth under the gravitational force. Here both velocity and acceleration are in the same direction because the ball is speeding up in a downward direction. Again from the above analysis, **when an object moving in –ve direction and speeding up, its acceleration is negative. **So in both halves of motion, the acceleration of a ball is negative.

**The circular motion of the ball attached to a string**

When a ball, attached to a massless string, performs circular motion different-different forces act on it. The massless and inextensible string provides the necessary centripetal force. We all know that centripetal force is always acts towards the center of a system. The direction of acceleration is along with centripetal force, i.e., always towards the system’s center in a circular motion. The direction of linear velocity is constantly changing in a circular motion. The radial acceleration of a circular motion is always negative.

The radial component of acceleration in circular motion is,

[latex]a_{r}=-r\omega ^{2}[/latex]

[latex]a_{r}[/latex] – radial acceleration

[latex]r[/latex] – radius of circle

[latex]\omega[/latex] – angular velocity

**Motion of Pendulum**

The pendulum’s motion is an oscillatory motion about its mean position, and to understand the nature of acceleration of a pendulum, let us break the pendulum’s motion into four cases.

**Case 1**– Pendulum moves in +ve x-direction from mean position to extreme position, and slowing down

In this case, restoring force provided by the component of gravity tries to pull the pendulum towards the mean position. Therefore the velocity of the pendulum decreases as it moves from the mean position to the extreme position. In that case acceleration of a pendulum is negative, and its direction is opposite to the pendulum’s motion.

**Case 2**– Pendulum moves from an extreme position to mean position (i.e., from right to left) in +x direction, and speeding up

In this case, the pendulum accelerates towards the mean position. So the motion and acceleration are in the same direction, but the direction of acceleration is from right to left, i.e., in a negative direction. Hence the acceleration of a pendulum is negative.

**Case 3:** Pendulum moves from mean position to the extreme position in –ve x-direction, and slowing down.

This case is almost similar to the first case, and only the direction is in –ve x-axis. In this case, acceleration is opposite to the motion and in +ve direction. Hence the acceleration is +ve for this case.

Case 4- Pendulum moves from extreme to the mean position (i.e., from left to right), and speeding up

Here the pendulum accelerates due to restoring force towards the mean position. Both velocities, as well as acceleration, are in the same direction. The direction of acceleration is in the +ve x-axis, so it is considered a positive acceleration.

**The motion of a car along a curved road**

The car performs circular motion when it moves along the curve road. Necessary centripetal force, required for the motion along a curved path, is provided by the friction between tire and road. Acceleration is directed towards the system’s center, and it is negative because its direction is from right to left i.e., in –ve x-direction.