Verified Solution • 2025–26 Session hard

QP 51-54: Q52: HC Verma: Concepts of Physics - Volume 1 Chapter-3 Rest and Motion Kinematics P 51-54: Q52 Solution: Six particles situated at the corner of a regular hexagon of…

⚠️ Common Exam Traps & Conceptual Pitfalls

Students frequently mistake the exterior angle ($60^circ$) for the angle between the velocity vector of the chased particle and the line joining the two particles, which is actually the interior angle ($120^circ$). This leads to using $cos(60^circ) = 1/2$ instead of $cos(120^circ) = -1/2$, incorrectly calculating the approach speed as $v/2$ instead of $3v/2$.

Six particles situated at the corner of a regular hexagon of side a move at a constant speed v. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.

Multiple Choice Options

  • A
  • B
  • C
  • D

Step-by-Step Verified Solution

The six particles will take a time t=2avt = \frac{2a}{v} to meet each other at the center of the hexagon.


Method 1: Relative Velocity of Approach

Due to symmetry, the particles will always form a shrinking regular hexagon at any instant tt.


  1. Consider any two adjacent particles, AA and BB, separated by distance r(t)r(t) along side ABAB.
  2. Particle AA moves along ABAB with speed vv directly towards BB.
  3. Particle BB moves along side BCBC with speed vv towards particle CC.
  4. The interior angle of a regular hexagon is 120∘120^\circ, which means the angle between side ABAB and side BCBC is 120∘120^\circ, making the angle between the velocity vector of BB and the line ABAB equal to 180∘−120∘=60∘180^\circ - 120^\circ = 60^\circ.

The component of BB's velocity directed opposite to AA's motion (along the line segment BABA) is vcos⁡(60∘)v \cos(60^\circ).


The rate at which the distance rr between AA and BB decreases (velocity of approach) is:


vapp=−drdt=v−vcos⁡(60∘)v_{\text{app}} = -\frac{dr}{dt} = v - v \cos(60^\circ)

Since cos⁡(60∘)=12\cos(60^\circ) = \frac{1}{2}:


vapp=v−v2=v2v_{\text{app}} = v - \frac{v}{2} = \frac{v}{2}

Since vappv_{\text{app}} remains constant throughout the motion, the time required to reduce the side length from aa to 00 is:


t=avapp=av2=2avt = \frac{a}{v_{\text{app}}} = \frac{a}{\frac{v}{2}} = \frac{2a}{v}

Method 2: Motion Towards the Center

For a regular hexagon of side length aa, the distance from any vertex to the center OO is R0=aR_0 = a.


  1. The line joining a vertex to the center forms an angle of 60∘60^\circ with each adjacent side.
  2. The component of the particle's velocity directed towards the center OO is:

vR=vcos⁡(60∘)=v2v_R = v \cos(60^\circ) = \frac{v}{2}

  1. Since the radial distance to the center decreases at a constant rate vR=v2v_R = \frac{v}{2}, the time taken to reach the center is:

t=R0vR=av2=2avt = \frac{R_0}{v_R} = \frac{a}{\frac{v}{2}} = \frac{2a}{v}

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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