Linear Acceleration: Definition, Formula, Examples, Units

If a fully loaded truck and a dirt bike are started simultaneously, the dirt bike can reach a speed of 100 km/h in a matter of seconds whereas the truck needs minutes to reach the same speed. The changes in the speed of both vehicles can be compared with the help of the term linear acceleration which indicates the change in velocity of the object per unit time.

Thus, in this article, we will go over the term linear acceleration in depth in order to clear up all doubts related to this topic.

What is Linear Acceleration?

The linear acceleration of an object traveling in a straight line is defined as the rate of change of its linear velocity with respect to time. It arises due to the change in magnitude of the velocity without changing the direction.

It may result from either an increase in velocity or a reduction in the velocity of a moving object. Let’s understand this with the help of the below example.

The Car-1 shown in the below figure is speeding up from rest to 4 m/s without changing its direction, thus it can be concluded that the car-1 possesses a linear acceleration.

linear acceleration of the car

The velocity of Car-2 is decreasing from 4 m/s to 0 m/s, thus the Car-2 also possesses a linear acceleration.

The Car-3 in the above figure is moving with a constant velocity of 2 m/s, thus the car-3 has zero linear acceleration.

The linear acceleration is given by,

a = `\frac{\text{Change in velocity}}{\text{Time}}`

The linear acceleration of an object also depends on its mass and the net force acting on it.

As per Newton’s second law of motion, the linear acceleration is directly proportional to the net force (`a\proptoF`) acting on the object and inversely proportional to its mass (`a\propto\frac{1}{m}`).

Thus the acceleration on the object increases with the increase in net force applied to the object. While for the same force applied, the object with less mass has higher acceleration than the object with higher mass.

Relation between mass and acceleration

What is linear deceleration?
Linear deceleration or linear retardation is the decrease in the velocity of the object with respect to time. The vector of deceleration has the opposite direction as that of the linear velocity of the object. The deceleration causes a decrease in the velocity of the object.

Types of linear acceleration:

Based on the extent of time taken for observation, the linear acceleration is of two types:-

A] Average linear acceleration:-

Average linear acceleration is the mean value of the acceleration calculated for a considerable duration. The average linear acceleration of any object is calculated as follows,

`a_{\text{average}} = \frac{V_{\text{Final}}-V_{\text{Initial}}}{t_{\text{Final}}-t_{\text{Initial}}}`

`a_{\text{average}} = \frac{\DeltaV}{\Deltat}`

B] Instantaneous linear acceleration:-

It is the linear acceleration of the object at a particular instant. It is generally calculated for the smaller interval dt.

It is given by,

`a_{\text{in}}=\lim_{\Deltat\to0}\frac{\DeltaV}{\Deltat}`

OR

`a_{\text{in}}=\frac{dV}{dt}`


Based on the direction of the linear acceleration, it can be classified as follows:-

A] Positive acceleration:-

The positive linear acceleration indicates that the acceleration vector of the object is in a positive direction. i.e. toward the right or upward. Let’s understand with the following example.

Positive acceleration in a horizontal direction

The biker shown in the above figure-A is moving in a positive direction (left to right). As she is speeding up, the acceleration also has the same direction as that of the velocity of the bike i.e. Positive direction. Therefore in this case, the acceleration is known as positive linear acceleration.

The biker shown in figure-B is moving in a negative direction (right to left). As he is slowing down, the direction of the acceleration is opposite to the direction of the velocity of the bike. Hence the acceleration vector is in a positive direction, and thus the bike possesses positive linear acceleration.

Positive acceleration in a vertical direction

 The above figure-C indicates the fighter jet speeding up in a vertical direction [+ve direction]. As the velocity of the jet is increasing, the acceleration and velocity have the same direction [+ve direction]. Thus in this case, the jet is moving with positive linear acceleration.

The above figure-D shows that the jet is moving in a downward direction. As the jet is slowing down, the direction of the acceleration is opposite to the direction of the velocity of the jet. Thus the jet has a positive linear acceleration.

B] Negative acceleration:-

The negative acceleration indicates that the acceleration vector of the object is in a negative direction i.e. toward left or downward. Let’s understand it with the following example.

negative acceleration in a horizontal direction

The biker in figure-A is moving in a positive direction (left to right). As she is speeding down, the acceleration has the opposite direction as that of the velocity of the bike. Therefore in this case the biker is experiencing negative acceleration.

While the biker in figure-B is moving in a negative direction (right to left). As he is speeding up, the velocity and acceleration of the biker are in the same direction (-ve direction). Thus the biker is experiencing negative linear acceleration.

The below figure-c and figure-d indicate the motion of the ball while moving up and moving down respectively.

Negative acceleration in a vertical direction

As shown in figure-c, when we throw a ball in an upward direction (+ve direction), the velocity of the ball goes on decreasing because of the gravitational acceleration (g).

Thus the acceleration vector is in opposite direction (-ve) as that of the velocity of the ball. Thus as the vector of the linear acceleration is pointing in a downward direction, the ball in figure-c is under negative linear acceleration.

The ball in figure-d is moving in a downward direction (-ve direction). The velocity of the ball is increasing due to the gravitational acceleration (g). Thus the direction of the acceleration is same as that of the direction of the velocity of the ball. hence the ball in figure-d is under negative linear acceleration.

C] Zero acceleration:-

Zero acceleration indicates that there is no change in the linear velocity of the object during a given time duration or it means that the object is moving with a constant velocity.

Linear acceleration formula:

The formula for the average linear acceleration is given by,

`a_{\text{average}}=\frac{\DeltaV}{\Deltat}=\frac{V_{\text{final}}-V_{\text{initial}}}{\Deltat}`

The formula of instantaneous linear acceleration is given by,

`a_{\text{in}}=\lim_{\Deltat\to0}\frac{\DeltaV}{\Deltat}=\frac{dV}{dt}`

Units:

As the acceleration is the ratio of the change in linear velocity to the time, thus in the SI system, the unit of the linear acceleration is given by,

`a = \frac{\text{Change in velocity}}{\text{Time}} = \frac{m/s}{s} = \frac{m}{s^{2}}`

Thus in the SI system of units, It is measured in terms of m/s².

Similarly, in the case of the FPS system, It is measured in terms of ft/s².

Linear acceleration solved examples:

1] If a car accelerates linearly from 0 km/h to 100 km/h in 12 seconds, what is its average acceleration?

Given:
`V_{\text{Initial}} = 0\ \text{Km/h} = 0\ \text{m/s}`
`V_{\text{Final}} = 100\ \text{Km/h} = \frac{100 \times 1000}{3600}= 27.77\ \text{m/s}`
`\Delta t = 12\ \text{seconds}`

Solution:-

The average linear acceleration of the car can be calculated as follows,

`a = \frac{ V_{\text{Final}}- V_{\text{Initial}}}{\Deltat}`

`a = \frac{27.77-0}{12}`

𝑎 = 2.314 m/s²


2] The bus initially running at the speed of 40 Km/h accelerates to attain a velocity of 60 Km/h within 10 seconds. Find the distance covered by the bus during the acceleration period.

Given:
`V_{\text{Initial}} = 40\ \text{Km/h} = \frac{40\times1000}{3600}= 11.11\ \text{m/s}`
`V_{\text{Final}} = 60\ \text{Km/h} = \frac{60\times1000}{3600} = 16.66\ \text{m/s}`
`\Deltat = 10\ \text{seconds}`

Solution:-

The average linear acceleration of the bus is given by,

`a = \frac{ V_{\text{Final}}- V_{\text{Initial}}}{\Deltat}`

`a = \frac{16.66-11.11}{10}`

𝑎 = 0.555 m/s²

By using Newton’s equation of motion, the displacement (S) of the bus is given by,

`S = ut + \frac{1}{2}a.t^{2}`

As, `u = V_{\text{Initial}}`, `t = \Deltat`, the above equation becomes,

`S = V_{\text{Initial}}.\Deltat+\frac{1}{2}a.\Deltat^{2}`

`S = (11.11\times10) + \frac{1}{2}(0.555\times10^{2})`

S = 138.85 m

This is the distance covered by the bus while accelerating from 40 Km/h to 60 Km/h.

FAQs:

  1. What are real examples of linear acceleration?

    The examples of linear acceleration are as follows,
    1] Falling of object due to gravitational acceleration.
    2] Speeding up the car in a straight direction.

  2. What is the dimensional formula for linear acceleration?

    The dimensional formula for the linear acceleration is [L¹M⁰T⁻²].

  3. How can linear acceleration be calculated?

    The linear acceleration is calculated by using the following equation,
    a = [V​​​​​🇫​​​​​🇮​​​​​🇳​​​​​🇦​​​​​🇱​​​​​​​​​​ – V🇮​​​​​🇳​​​​​🇮​​​​​🇹​​​​​🇮​​​​​🇦​​​​​🇱​​​​​] /Δt.

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