Verified Solution • 2025–26 Session moderate

QP 50-51: Q2: HC Verma: Concepts of Physics - Volume 1 Chapter-3 Rest and Motion Kinematics P 50-51: Q2 Solution: A particle moves along the X-axis as x = u(t − 2 s) + a(t − 2…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students often mistakenly identify the coefficient $a$ in the position equation $x = u(t - t_0) + a(t - t_0)^2$ as the acceleration, ignoring that the second derivative with respect to time yields $2a$. Always differentiate the full time-shifted expression to determine the true acceleration and initial velocity at $t=0$.

A particle moves along the X-axis as x = u(t − 2 s) + a(t − 2 s)².

Multiple Choice Options

  • A the initial velocity of the particle is u
  • B the acceleration of the particle is a
  • C the acceleration of the particle is 2a
  • D at t = 2 s particle is at the origin.

Step-by-Step Verified Solution

Given the position equation x=u(t−2)+a(t−2)2x = u(t - 2) + a(t - 2)^2, we can evaluate each statement by finding the velocity and acceleration as functions of time.

1. Velocity Analysis

Differentiating position xx with respect to time tt:

v=dxdt=ddt[u(t−2)+a(t−2)2]=u+2a(t−2)v = \frac{dx}{dt} = \frac{d}{dt}\left[u(t - 2) + a(t - 2)^2\right] = u + 2a(t - 2)

  1. Initial velocity refers to velocity at t=0 st = 0\text{ s}:
  2. v(0)=u+2a(0−2)=u−4av(0) = u + 2a(0 - 2) = u - 4a
  3. (Statement "the initial velocity of the particle is uu" is incorrect; uu is the velocity at t=2 st = 2\text{ s}.)

2. Acceleration Analysis

Differentiating velocity vv with respect to time tt:

A=dvdt=ddt[u+2a(t−2)]=2aA = \frac{dv}{dt} = \frac{d}{dt}\left[u + 2a(t - 2)\right] = 2a

  1. Statement "the acceleration of the particle is aa" is incorrect.
  2. Statement "the acceleration of the particle is 2a2a" is correct

3. Position at t=2 st = 2\text{ s}

Substituting t=2 st = 2\text{ s} into the position equation:

x(2)=u(2−2)+a(2−2)2=0x(2) = u(2 - 2) + a(2 - 2)^2 = 0

  1. Statement "at t=2 st = 2\text{ s} particle is at the origin" is correct.

Summary of Statements

  1. Incorrect: The initial velocity is uu.
  2. Incorrect: The acceleration is aa.
  3. Correct: The acceleration of the particle is 2a2a.
  4. Correct: At t=2 st = 2\text{ s}, the particle is at the origin.


All Questions in this Chapter

QP 49-50: Q1 QP 49-50: Q2 QP 49-50: Q3 QP 49-50: Q4 QP 49-50: Q5 QP 49-50: Q6 QP 49-50: Q7 QP 49-50: Q8 QP 49-50: Q9 QP 49-50: Q10 QP 49-50: Q11 QP 49-50: Q12 QP 49-50: Q13 QP 50-51: Q1 QP 50-51: Q2 QP 50-51: Q3 QP 50-51: Q4 QP 50-51: Q5 QP 50-51: Q6 QP 50-51: Q7 QP 50-51: Q8 QP 50-51: Q9 QP 50-51: Q10 QP 51-54: Q1 QP 51-54: Q2 QP 51-54: Q3.1 QP 51-54: Q3.2 QP 51-54: Q3.3 QP 51-54: Q3.4 QP 51-54: Q4.1 QP 51-54: Q4.2 QP 51-54: Q5 QP 51-54: Q6 QP 51-54: Q7 QP 51-54: Q8.1 QP 51-54: Q8.2 QP 51-54: Q8.3 QP 51-54: Q9.1 QP 51-54: Q9.2 QP 51-54: Q10 QP 51-54: Q11 QP 51-54: Q12 QP 51-54: Q13 QP 51-54: Q14 QP 51-54: Q15.1 QP 51-54: Q15.2 QP 51-54: Q15.3 QP 51-54: Q16 QP 51-54: Q17 QP 51-54: Q18.1 QP 51-54: Q18.2 QP 51-54: Q19 QP 51-54: Q20 QP 51-54: Q21 QP 51-54: Q22 QP 51-54: Q23.1 QP 51-54: Q23.2 QP 51-54: Q23.3 QP 51-54: Q24 QP 51-54: Q25.1 QP 51-54: Q25.2 QP 51-54: Q25.3 QP 51-54: Q26 QP 51-54: Q27 QP 51-54: Q28 QP 51-54: Q29 QP 51-54: Q30 QP 51-54: Q31 QP 51-54: Q32.1 QP 51-54: Q32.2 QP 51-54: Q32.3 QP 51-54: Q33.1 QP 51-54: Q33.2 QP 51-54: Q34 QP 51-54: Q35 QP 51-54: Q36 QP 51-54: Q37 QP 51-54: Q38 QP 51-54: Q39 QP 51-54: Q40 QP 51-54: Q41 QP 51-54: Q42 QP 51-54: Q43.1 QP 51-54: Q43.2 QP 51-54: Q44 QP 51-54: Q45 QP 51-54: Q46.1 QP 51-54: Q46.2 QP 51-54: Q47.1 QP 51-54: Q47.2 QP 51-54: Q48 QP 51-54: Q49.1 QP 51-54: Q49.2 QP 51-54: Q50 QP 51-54: Q51 QP 51-54: Q52 QPage 48-49: Q1 QPage 48-49: Q2 QPage 48-49: Q3 QPage 48-49: Q4 QPage 48-49: Q5 QPage 48-49: Q6 QPage 48-49: Q7 QPage 48-49: Q8 QPage 48-49: Q9 QPage 48-49: Q10.1 QPage 48-49: Q10.2 QPage 48-49: Q10.3 QPage 48-49: Q11 QPage 48-49: Q12 QPage 48-49: Q13 QPage 48-49: Q14

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