Derive Formula v^2 – u^2 = 2as, Equation of Uniformly Accelerated Motion MCQs, FAQs, Q&A, Worksheet

ANAND CLASSES study material and notes to learn the third equation of motion $v^2 – u^2 = 2as$ with a step-by-step derivation, conceptual explanations, solved examples, and practice problems for JEE, NEET, and CBSE Class 11 physics.

The third equation of motion is an essential kinematic equation that describes the relationship between the final velocity $v$, initial velocity $u$, acceleration $a$, and displacement $s$ of an object moving under uniform acceleration. It eliminates the time variable, making it useful for solving problems where time is not given.

Derivation of the Third Equation of Motion($v^2 – u^2 = 2as$)

The third equation of motion can be derived using the first two equations of motion:

Step 1: Recall the Second Equation of Motion

The second equation of motion is given by: $$s = ut + \frac{1}{2}at^2$$

Step 2: Express Time tt Using the First Equation of Motion

The first equation of motion is: $$v = u + at$$

Rearranging for $t$ : $$t = \frac{v – u}{a}$$

Step 3: Substituting tt in the Second Equation

Substituting $$t = \frac{v – u}{a}$$

into

$$s = ut + \frac{1}{2}at^2 $$

we get,

$$s = u \left( \frac{v – u}{a} \right) + \frac{1}{2} a \left( \frac{(v – u)^2}{a^2} \right)$$

Expanding each term,

$$s = \frac{u(v – u)}{a} + \frac{(v^2 + u^2 – 2uv)}{2a}$$

$$s = \frac{2uv – 2u^2 + v^2 – 2uv + u^2}{2a}$$

Simplifying,

$$2as = v^2 – u^2 $$

or

$$ v^2 – u^2 = 2as$$

Thus, the third equation of motion is derived.

Significance of the Third Equation of Motion

• It relates velocity and displacement without involving time.
• It is useful when time is not given in the problem statement.
• It helps in analyzing motion under uniform acceleration.

Important Points to Remember

• If a body starts from rest, $u = 0$.
• If a body comes to rest, $v = 0$.
• If a body moves with uniform velocity, $a = 0$.

Question-Answer Format for Exams (JEE, NEET, CBSE)

Q1. What does the third equation of motion signify?

A1. It signifies the relationship between velocity, acceleration, and displacement without involving time. It is particularly useful in cases where the time of travel is unknown.

Q2. Can the third equation of motion be used for non-uniform acceleration?

A2. No, it is derived under the assumption of constant acceleration. If acceleration is non-uniform, the equation does not hold.

MCQs with Explanation

Q1. Which of the following equations represents the third equation of motion?
(a) $v = u + at$
(b) $s = ut + \frac{1}{2}at^2$
(c) $v^2 = u^2 + 2as$
(d) $a = \frac{v – u}{t}$

Answer: (c) $v^2 = u^2 + 2as$
Explanation: This equation is derived using the first two kinematic equations and does not include time, making it useful in problems where time is unknown.

Do You Know?

• The third equation of motion is also called the velocity-displacement equation.
• It plays a crucial role in understanding free fall motion, where $a = g$ (acceleration due to gravity).
• It is widely used in physics and engineering to analyze motion in one dimension.

Worksheet(Solve the Following Problems)

1. A car accelerates uniformly from 10 m/s to 30 m/s over a distance of 200 m. Find the acceleration.
2. A ball is dropped from a height of 100 m. Find its final velocity before hitting the ground. (Take $g = 9.8$ m/s²)
3. A cyclist moving at 5 m/s accelerates at 2 m/s² for a distance of 25 m. Find the final velocity.

Test Paper (Total Marks: 10)

1. Derive the third equation of motion. (4 marks)
2. A train starts from rest and moves with an acceleration of 2 m/s². Find its velocity after traveling 100 m. (3 marks)
3. A car moving with an initial velocity of 20 m/s comes to rest after traveling 80 m. Find the acceleration. (3 marks)

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