ANAND CLASSES study material and notes to learn the fundamentals of acceleration in mechanics with easy explanations, formulas, and examples. Perfect for JEE, NEET, and CBSE Class 11 students.
What is Acceleration?
When the velocity of a body increases over time, it is said to be accelerating.
Example of Acceleration
Imagine a car starting from rest (i.e., initial velocity = 0). If its velocity increases steadily and reaches 10 m/s in 5 seconds, we calculate the acceleration as follows:
Step 1: Find the Change in Velocity
$$\text{Change in velocity} =\text{Final velocity} -\text{Initial velocity}$$
$$\text{Change in velocity}=10 – 0 = 10 \text{ m/s}$$
Step 2: Find Acceleration
Acceleration is given by the formula:
$$\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time taken for change}}$$
$$\text{Acceleration} = \frac{10}{5} = 2 \text{ m/s}^2$$
Thus, the car has an acceleration of 2 m/s², meaning its velocity increases by 2 meters per second every second.
Mathematical Definition of Acceleration
Acceleration (a) is defined as the rate of change of velocity with respect to time: $$a = \frac{v – u}{t}$$
Where:
- $a$ = Acceleration (m/s²)
- $v$ = Final velocity (m/s)
- $u$ = Initial velocity (m/s)
- $t$ = Time taken (s)
Key Points to Remember
✅ Acceleration is positive when velocity increases.
✅ Acceleration is negative (deceleration) when velocity decreases.
✅ SI unit of acceleration: m/s².
Definition of Acceleration
Acceleration is the rate at which the velocity of an object changes with time. It is a vector quantity, meaning it has both magnitude and direction. Mathematically, acceleration is given by:
$$a = \frac{dv}{dt}$$
where:
- $a$ = acceleration (m/s²)
- $v$ = velocity (m/s)
- $t$ = time (s)
Key Characteristics of Acceleration:
- Vector Quantity: Acceleration has both magnitude and direction.
- Direction of Acceleration: The direction of acceleration is always the same as the direction of the change in velocity (not necessarily in the direction of velocity itself).
- Ways Velocity Can Change:
- Change in magnitude only (speeding up or slowing down).
- Change in direction only (e.g., uniform circular motion).
- Change in both magnitude and direction (e.g., projectile motion).
Types of Acceleration
1. Uniform Acceleration
A body is said to have uniform acceleration if its acceleration remains constant over time. This means both magnitude and direction of acceleration do not change.
Example: Free fall under gravity (where acceleration = 9.8 m/s²).
2. Non-Uniform Acceleration
A body is said to have non-uniform acceleration if either the magnitude or direction (or both) of acceleration changes during motion.
Example: A car moving in a busy road with variable speeds.
3. Average Acceleration
The average acceleration over a time interval is the total change in velocity divided by the total time taken: $$a_{av} = \frac{{v_2} – {v_1}}{\Delta t} $$
where:
- $v_1$ = initial velocity
- $v_2$ = final velocity
- $\Delta t$ = time interval
4. Instantaneous Acceleration
Instantaneous acceleration is the acceleration at a particular moment in time and is defined as: $$a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{d v}{dt}$$
Acceleration in Different Cases
| Case | Acceleration Type | Example |
|---|---|---|
| Only direction changes | Acceleration is perpendicular to velocity | Uniform circular motion |
| Only magnitude changes | Acceleration is parallel or anti-parallel to velocity | Free fall under gravity |
| Both magnitude and direction change | Acceleration has both components | Projectile motion |
Newton’s Second Law and Acceleration
Newton’s Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass: $$F = m a$$
where:
- $F$ = force (N)
- $m$ = mass (kg)
- $a$ = acceleration (m/s²)
Example: If a 10 N force is applied to a 2 kg object, its acceleration is: $$a = \frac{F}{m} = \frac{10}{2} = 5 \text{ m/s}^2$$
FAQs
Q1: Can acceleration be negative?
✅ Yes, negative acceleration (retardation) occurs when velocity decreases over time. Example: A car slowing down.
Q2: What is the acceleration due to gravity?
✅ The acceleration due to gravity on Earth is 9.8 m/s².
Q3: How does acceleration relate to velocity and displacement?
✅ Acceleration is the second derivative of displacement and the first derivative of velocity: $$a = \frac{dv}{dt} = \frac{d^2 x}{dt^2}$$
MCQs with Answers
Q1: What is the SI unit of acceleration?
- (A) m/s
- (B) m/s² ✅
- (C) cm/s²
- (D) N/kg
Q2: A car accelerates uniformly from rest to 20 m/s in 4 seconds. What is its acceleration?
A : Acceleration is given by :
$$a = \frac{v – u}{t} = \frac{20 – 0}{4} = 5 \text{ m/s}^2$$
✅ Correct Answer: 5 m/s²
Conceptual Questions with Answers
Q1: Can an object have zero velocity but non-zero acceleration?
✅ Yes, at the highest point of a projectile’s motion, velocity is zero, but acceleration due to gravity still acts downward.
Q2: Why does a ball thrown upwards slow down before stopping?
✅ The acceleration due to gravity provides a negative acceleration, reducing velocity until it momentarily stops before falling back.
Do You Know?
- The acceleration due to gravity on the Moon is 1.63 m/s².
- Negative acceleration is also called retardation or deceleration.
- Circular motion always involves centripetal acceleration, directed toward the center.
Worksheet
- Define acceleration and state its SI unit. (2 marks)
- Differentiate between uniform and non-uniform acceleration with examples. (3 marks)
- A car speeds up from 10 m/s to 30 m/s in 5 seconds. Find its acceleration. (2 marks)
- What is instantaneous acceleration? Explain with an example. (3 marks)
- A ball is dropped from a height. Find its velocity after 3 seconds. (4 marks)
Test Paper (20 Marks)
Section A: Short Answer (2 marks each)
- Define acceleration and give its SI unit.
- Explain uniform and non-uniform acceleration with examples.
- What is retardation? Give an example.
- How is acceleration related to force?
Section B: Numerical Problems (4 marks each)
- A vehicle accelerates from 5 m/s to 25 m/s in 10 seconds. Calculate acceleration.
- A force of 50 N acts on a 10 kg mass. Find acceleration.
- A body falls freely under gravity for 4 seconds. Find its final velocity and distance covered.
Quick Revision Points
- Acceleration = Rate of change of velocity.
- SI Unit: m/s².
- Types: Uniform, Non-uniform, Average, Instantaneous.
- Newton’s Second Law: F = ma.
- Gravity causes 9.8 m/s² acceleration on Earth.
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