3.1 Acceleration | Translatory Motion | Physics 11

Acceleration is the measure of how quickly an object's velocity changes over time. Since velocity is a Scalar And Vector Quantities→ (having both magnitude and direction), acceleration occurs whenever an object speeds up, slows down, or changes its direction of motion. It is a fundamental concept in kinematics, the branch of physics that describes motion without considering its causes.

Definition and Formula

Acceleration is defined as the rate of change of velocity with respect to time.

Mathematically, average acceleration is expressed as:

$a_{a v g} = \frac{Δ v}{Δ t} = \frac{v _{f} - v _{i}}{t _{f} - t _{i}}$

Where:

$v_{f}$ is the final velocity
$v_{i}$ is the initial velocity
$Δ t$ is the time interval
Vector Quantity: Acceleration has both magnitude and direction, making it a vector quantity.
SI Unit: The standard unit for acceleration is meters per second squared ( $m/s^{2}$ ). In terms of base SI units, this is expressed as $m s^{- 2}$ .
Dimensions: The Dimensions→ of acceleration are $[L T^{- 2}]$ .

Types of Acceleration

Average Acceleration: The total change in velocity divided by the total time interval. It gives the overall change in velocity over a period but does not describe motion at any specific instant.

Instantaneous Acceleration: The acceleration of an object at a specific instant in time. It is obtained by taking the limit of average acceleration as the time interval approaches zero:

$a_{in s t} = lim_{Δ t \to 0} \frac{Δ v}{Δ t}$

Uniform Acceleration: An object experiences uniform (or constant) acceleration when its velocity changes by equal amounts in equal intervals of time. Under uniform acceleration, instantaneous acceleration equals average acceleration at all times.

Variable Acceleration: An object has variable acceleration when its velocity changes by unequal amounts in equal intervals of time.

Direction of Acceleration

The direction of the acceleration vector determines whether an object is speeding up or slowing down.

Speeding Up: If an object is speeding up, its acceleration is in the same direction as its velocity.
Slowing Down: If an object is slowing down, its acceleration is in the opposite direction to its velocity. This type of acceleration is called deceleration or retardation.

Positive and Negative Acceleration

The sign of acceleration depends on the chosen coordinate system. Negative acceleration does not always mean the object is slowing down.

Scenario	Velocity	Acceleration	Result
Object moving in positive direction and speeding up	Positive	Positive	Speeding up
Object moving in positive direction and slowing down	Positive	Negative	Slowing down (Deceleration)
Object moving in negative direction and speeding up	Negative	Negative	Speeding up
Object moving in negative direction and slowing down	Negative	Positive	Slowing down (Deceleration)

Graphical Representation

The motion of an object can be analyzed using graphs:

Velocity-Time (v-t) Graph: The slope of the velocity-time graph represents the acceleration of the object: $a = \frac{Δ v}{Δ t}$
Acceleration-Time (a-t) Graph: The area under an acceleration-time graph represents the change in velocity: $Δ v = a \cdot Δ t$

Equations of Motion for Uniform Acceleration

When acceleration is constant (uniform), the following three equations of motion can be derived and applied. Here $v_{i}$ is initial velocity, $v_{f}$ is final velocity, $a$ is constant acceleration, $s$ is displacement, and $t$ is time:

$v_{f} = v_{i} + a t ...(1)$

$s = v_{i} t + \frac{1}{2} a t^{2} ...(2)$

$v_{f}^{2} = v_{i}^{2} + 2 a s ...(3)$

Important: These equations are valid only for uniform (constant) acceleration in a straight line.

Worked Examples

Example 1 — Zero velocity with non-zero acceleration

Question: Can an object have zero velocity but non-zero acceleration?

Answer: Yes. At the very peak of its trajectory, a ball thrown straight upward has instantaneous velocity equal to zero, but it is still accelerating downward due to gravity (approximately $9.8 m s^{- 2}$ ). This demonstrates that acceleration and velocity are independent quantities.

Example 2 — Applying equations of motion

Question: A car starts from rest and accelerates uniformly at $3 m s^{- 2}$ . Find (a) its velocity after $4 s$ , and (b) the distance covered in that time.

Solution:

Given: $v_{i} = 0$ , $a = 3 m s^{- 2}$ , $t = 4 s$

(a) Using equation (1): $v_{f} = v_{i} + a t = 0 + (3) (4) = 12 m s^{- 1}$

(b) Using equation (2): $s = v_{i} t + \frac{1}{2} a t^{2} = 0 + \frac{1}{2} (3) (4)^{2} = 24 m$

Definition and Formula

Acceleration is defined as the rate of change of velocity with respect to time.

Mathematically, average acceleration is expressed as:

$a_{a v g} = \frac{Δ v}{Δ t} = \frac{v _{f} - v _{i}}{t _{f} - t _{i}}$

Where:

$v_{f}$ is the final velocity
$v_{i}$ is the initial velocity
$Δ t$ is the time interval
Vector Quantity: Acceleration has both magnitude and direction, making it a vector quantity.
SI Unit: The standard unit for acceleration is meters per second squared ( $m/s^{2}$ ). In terms of base SI units, this is expressed as $m s^{- 2}$ .
Dimensions: The Dimensions→ of acceleration are $[L T^{- 2}]$ .

Types of Acceleration

Average Acceleration: The total change in velocity divided by the total time interval. It gives the overall change in velocity over a period but does not describe motion at any specific instant.

Instantaneous Acceleration: The acceleration of an object at a specific instant in time. It is obtained by taking the limit of average acceleration as the time interval approaches zero:

$a_{in s t} = lim_{Δ t \to 0} \frac{Δ v}{Δ t}$

Variable Acceleration: An object has variable acceleration when its velocity changes by unequal amounts in equal intervals of time.

Direction of Acceleration

The direction of the acceleration vector determines whether an object is speeding up or slowing down.

Speeding Up: If an object is speeding up, its acceleration is in the same direction as its velocity.
Slowing Down: If an object is slowing down, its acceleration is in the opposite direction to its velocity. This type of acceleration is called deceleration or retardation.

Positive and Negative Acceleration

The sign of acceleration depends on the chosen coordinate system. Negative acceleration does not always mean the object is slowing down.

Scenario	Velocity	Acceleration	Result
Object moving in positive direction and speeding up	Positive	Positive	Speeding up
Object moving in positive direction and slowing down	Positive	Negative	Slowing down (Deceleration)
Object moving in negative direction and speeding up	Negative	Negative	Speeding up
Object moving in negative direction and slowing down	Negative	Positive	Slowing down (Deceleration)

Graphical Representation

The motion of an object can be analyzed using graphs:

Velocity-Time (v-t) Graph: The slope of the velocity-time graph represents the acceleration of the object: $a = \frac{Δ v}{Δ t}$
Acceleration-Time (a-t) Graph: The area under an acceleration-time graph represents the change in velocity: $Δ v = a \cdot Δ t$

Equations of Motion for Uniform Acceleration

$v_{f} = v_{i} + a t ...(1)$

$s = v_{i} t + \frac{1}{2} a t^{2} ...(2)$

$v_{f}^{2} = v_{i}^{2} + 2 a s ...(3)$

Important: These equations are valid only for uniform (constant) acceleration in a straight line.

Worked Examples

Example 1 — Zero velocity with non-zero acceleration

Question: Can an object have zero velocity but non-zero acceleration?

Example 2 — Applying equations of motion

Question: A car starts from rest and accelerates uniformly at $3 m s^{- 2}$ . Find (a) its velocity after $4 s$ , and (b) the distance covered in that time.

Solution:

Given: $v_{i} = 0$ , $a = 3 m s^{- 2}$ , $t = 4 s$

(a) Using equation (1): $v_{f} = v_{i} + a t = 0 + (3) (4) = 12 m s^{- 1}$

(b) Using equation (2): $s = v_{i} t + \frac{1}{2} a t^{2} = 0 + \frac{1}{2} (3) (4)^{2} = 24 m$