Derive Formula v = u + at, Equation of Uniformly Accelerated Motion, MCQs, FAQs, Q&A, Worksheet

Motion with uniform acceleration is a fundamental concept in physics, particularly in mechanics. The equations of motion describe the relationship between initial velocity, final velocity, acceleration, time, and displacement. These equations are essential for solving problems in JEE, NEET, and CBSE Board Class 11 exams.

Derive $v = u + at$, First Equation of Uniformly Accelerated Motion

Statement : The first equation of motion is:

$$v = u + at$$

where:

• $v$ = final velocity of the body
• $u$ = initial velocity of the body
• $a$ = acceleration
• $t$ = time taken

This equation gives the velocity acquired by a body in time tt when it undergoes uniform acceleration.

Derivation of First Equation of Motion

Consider a body moving with an initial velocity uu. Suppose it is subjected to a uniform acceleration $a$ such that after time $t$, its final velocity becomes $v$. From the definition of acceleration:

$$\text{Acceleration} = \frac{\text{Change in velocity}}{\text{Time taken}}$$

or, $$a = \frac{v – u}{t}$$

Multiplying both sides by $t$, we get: $$at = v – u$$

Rearranging we get, $$v = u + at$$

Significance of First Equation of Motion

• It helps determine the velocity of a body at any given time during uniformly accelerated motion.
• If three values are known, the fourth can be easily calculated.
• The equation applies to retardation cases by using negative acceleration.

Question-Answer Format for JEE, NEET & CBSE Board Class 11

Conceptual Questions with Answers

Q1. What does the first equation of motion represent?
A: The first equation of motion $v = u + at$ represents the velocity acquired by a body in time tt when it undergoes uniform acceleration.

Q2. If a body starts from rest, how does the first equation simplify?
A: When a body starts from rest, $u = 0$, so the equation simplifies to $v = at$

Q3. What happens if acceleration aa is negative?
A: If aa is negative, the body undergoes retardation or deceleration, reducing its velocity over time.

Multiple Choice Questions (MCQs) with Explanations

Q1. The first equation of motion is used to determine:

A) Velocity of a body after time $t$
B) Displacement of a body
C) Acceleration of a body
D) Distance traveled in time $t$
Answer: A) Velocity of a body after time $t$
Explanation: The first equation of motion directly relates velocity with time and acceleration.

Q2. A body has an initial velocity of 10 m/s and accelerates at 22 m/s². What will be its velocity after 5 seconds?
A) 20 m/s
B) 15 m/s
C) 25 m/s
D) 30 m/s
Answer: C) 25 m/s
Explanation: Using $v = u + at$, we get $v = 10 + (2 \times 5)$ = 25 m/s.

Do You Know?

• The first equation of motion is derived from the definition of acceleration.
• It applies to both forward motion (positive acceleration) and backward motion (negative acceleration or retardation).
• This equation is a special case of kinematic equations and is widely used in mechanics.

Worksheet: First Equation of Motion

Solve the following problems:

1. A car starts with an initial velocity of 5 m/s and accelerates uniformly at 3 m/s² for 4 seconds. Find its final velocity.
2. A cyclist moving at 12 m/s applies brakes, causing a uniform deceleration of 22 m/s². How long will it take to stop?
3. A ball is dropped from a height and gains a velocity of 20 m/s in 2 seconds. What is the acceleration acting on it?

Test Paper: Equations of Motion

1. State and derive the first equation of motion. (5 marks)
2. A body starts from rest and accelerates uniformly at 4 m/s². What will be its velocity after 3 seconds? (3 marks)
3. A train moving at 30 m/s slows down with a uniform retardation of 5 m/s². How long will it take to stop? (4 marks)
4. If the acceleration of a moving body is zero, what can you conclude about its velocity? (3 marks)
5. A ball rolling on a surface at 8 m/s slows down with an acceleration of 2 m/s². Find the time taken to stop. (5 marks)

Quick Revision Points

• The first equation of motion: $v = u + at$
• Used to find final velocity when acceleration and time are known.
• Derived from the definition of acceleration.
• Works for both acceleration and retardation cases.
• If any three values ($v, u, a, t$) are known, the fourth can be found.

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