Dimensional Formula of Acceleration

Acceleration is the rate of change of an object’s velocity. Acceleration can be either constant or varying over time. In the case of constant acceleration, its value is a combination of the net change in velocity and total time over which this occurred. It is also called average acceleration.

Acceleration

In terms of both speed and direction, acceleration is the rate at which velocity changes with time. A point or object is said to be accelerated if it moves faster or slower in a straight line. Because the direction of motion on a circle is continually changing, the motion is accelerated even if the speed remains constant. Both effects contribute to an acceleration in all other types of motion.

Acceleration is a vector quantity since it has both a magnitude and a direction. Velocity is a vector quantity as well. It is defined as the change in velocity vector over a time interval divided by the time interval.

Through examples, such as a ball falling off from a rooftop and stopping a bus at a bus stop, it can be understood that acceleration occurs whenever the speed or direction of a moving object changes.

Formula of Acceleration

a = Δv/Δt

Where:

a is acceleration

∆v is a change in velocity

∆t is the time taken

• The SI unit for acceleration is m/s² (metre per second sq.)

The equations of motion are given as:

• v = u + at
• v² = u² + 2as
• s = ut + ½at²

Where:

v stands for velocity

u stands for initial velocity

a is acceleration

t stands for the time taken

s is distance travelled

The dimensional formula of acceleration

Any physical quantity’s dimensional formula is the formula that describes how and which of the base quantities are included in that quantity.

The dimensional formula of acceleration due to gravity is the same as acceleration and is written as [M⁰ L¹ T -2].

Where:

M is mass

L is length

T is time

Derivation of dimensional formula

Force = Mass × Acceleration due to gravity

Acceleration due to gravity = Force × (Mass)-1 …. (1)

Dimensional formula of mass = [M¹ L⁰ T⁰] ….(2)

And, dimensions of force = [M¹ L² T -2] ….(3)

On substituting equation (2) and (3) in equation (1), we get:

• g = [M¹ L¹ T -2] × [M¹ L⁰ T⁰]-1 = [M0 L¹ T -2]

Therefore, the dimensional formula of acceleration due to gravity is the same as acceleration and is written as [M⁰ L¹ T -2].

Types of Acceleration

• Uniform acceleration

If equal changes in velocity occur in equal intervals of time, regardless of how small these intervals are, a body is said to be travelling with uniform acceleration.

Example of uniform acceleration: free-falling of an object.

• Average/non-uniform acceleration

A particle is said to move with non-uniform acceleration if its acceleration is not uniform throughout a time interval. Average acceleration is defined as the ratio of velocity change to time.

Example of average acceleration: a car stopping at a traffic signal.

Average acceleration is denoted as ā for a given interval of time.

ā = (v2 – v1)/(t2 – t1) = ∆v/∆t

Where:

v2 and v1 are instantaneous velocities, and t2 and t1 and ā is average acceleration.

Instantaneous acceleration

Instantaneous acceleration is the acceleration of an item at any instant.

Direction of acceleration vector:

Because acceleration is a vector quantity, it has a corresponding direction. Two factors determine the acceleration vector’s direction:

• whether the thing is travelling in the + or – direction 
• whether the object is speeding up or slowing down

Conclusion

One of the most important aspects of motion is acceleration. The mathematical relationships between the parameters of displacement (d), velocity (v), and acceleration (a) are all quite close. The rate of velocity change to time is known as acceleration.

The following is the general principle for determining acceleration:

• The acceleration of an object slowing down is in the opposite direction of its velocity.
• This principle can be used to identify whether an object’s acceleration is positive or negative, right or left, up or down, etc.