Question

Which of the following is an equation of linear motion?

• A. v = u + at
• B. s = ut + 1/2 at²
• C. v² = u² + 2as
• D. all of these

Hint:

These equations are only valid when acceleration ($a$) is constant. If acceleration changes over time, you must use calculus (integration/differentiation) to find the motion parameters.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The equations of linear motion (also known as kinematic equations) describe the behavior of an object moving in a straight line under constant acceleration.

Step 2: Detailed Explanation:
There are three fundamental equations for uniformly accelerated motion:
• First Equation: $v = u + at$ (relates velocity and time).
• Second Equation: $s = ut + \frac{1}{2}at^2$ (relates displacement and time).
• Third Equation: $v^2 = u^2 + 2as$ (relates velocity and displacement). Since all options (A), (B), and (C) are valid kinematic equations, (D) is the correct choice.

Step 3: Final Answer:
All the provided options are equations of linear motion.

Answer 2

This question asks which of the listed relations is a genuine equation of linear motion under constant acceleration. Rather than simply naming the three equations, we can derive each one directly from the definitions of velocity and acceleration to confirm whether it legitimately belongs to this family.

1. \( v = u + at \): By definition, acceleration is the rate of change of velocity, \( a = \frac{dv}{dt} \). For constant \( a \), integrating both sides with respect to time from \( 0 \) to \( t \) gives \( v - u = at \), that is, \( v = u + at \). This is a valid, directly derivable equation of motion.
2. \( s = ut + \frac{1}{2}at^2 \): Velocity is the rate of change of displacement, \( v = \frac{ds}{dt} \). Substituting \( v = u + at \) and integrating with respect to time from \( 0 \) to \( t \) gives \( s = ut + \frac{1}{2}at^2 \). This too follows directly from the definitions, so it is valid.
3. \( v^2 = u^2 + 2as \): Eliminating time \( t \) between the first two equations (solving \( t = \frac{v-u}{a} \) from the first and substituting into the second) reduces the pair to \( v^2 = u^2 + 2as \). Since it is derived purely algebraically from the other two valid equations, it must also be valid.
4. All of these: Because each of the three equations above independently follows from the basic definitions of velocity and acceleration under constant acceleration, none of them can be excluded, so this option correctly covers all three.

Since every one of the three named relations is a properly derivable equation of linear motion, no single one of them can be picked over the others.

Therefore, the correct answer is all of these.