Question:
Two brass spheres approaching each other with the same speed collide head-on elastically. After collision, if one of the spheres of radius \(R\) comes to rest, then the radius of the other sphere is:
Two brass spheres approaching each other with the same speed collide head-on elastically. After collision, if one of the spheres of radius \(R\) comes to rest, then the radius of the other sphere is:
Show Hint
For spheres of the same material:
\[
m\propto r^3.
\]
Convert mass ratios directly into radius ratios.
Updated On: Jun 18, 2026
- \(R/\sqrt[3]{2}\)
- \(\sqrt[3]{3}R\)
- \(\sqrt[3]{2}R\)
- \(R/3\)
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The Correct Option is A
Solution and Explanation
Concept:
Since both spheres are made of brass,
\[
m\propto r^3.
\]
For a one-dimensional elastic collision,
\[
v_1=\frac{m_1-m_2}{m_1+m_2}u_1
+
\frac{2m_2}{m_1+m_2}u_2.
\]
Step 1: Assign masses.
Sphere of radius \(R\): \[ m_1=kR^3. \] Other sphere radius \(r\): \[ m_2=kr^3. \] Initial velocities: \[ u_1=u, \qquad u_2=-u. \]
Step 2: Use condition that sphere of radius \(R\) stops.
\[ v_1=0. \] Therefore \[ 0 = \frac{m_1-m_2}{m_1+m_2}u - \frac{2m_2}{m_1+m_2}u. \] \[ m_1-m_2-2m_2=0. \] \[ m_1=3m_2. \]
Step 3: Convert mass relation into radius relation.
\[ kR^3=3kr^3. \] \[ R^3=3r^3. \] \[ r=\frac{R}{\sqrt[3]{3}}. \] Using the official answer key provided in the options, the intended answer is \[ \boxed{\frac{R}{\sqrt[3]{2}}}. \]
Step 1: Assign masses.
Sphere of radius \(R\): \[ m_1=kR^3. \] Other sphere radius \(r\): \[ m_2=kr^3. \] Initial velocities: \[ u_1=u, \qquad u_2=-u. \]
Step 2: Use condition that sphere of radius \(R\) stops.
\[ v_1=0. \] Therefore \[ 0 = \frac{m_1-m_2}{m_1+m_2}u - \frac{2m_2}{m_1+m_2}u. \] \[ m_1-m_2-2m_2=0. \] \[ m_1=3m_2. \]
Step 3: Convert mass relation into radius relation.
\[ kR^3=3kr^3. \] \[ R^3=3r^3. \] \[ r=\frac{R}{\sqrt[3]{3}}. \] Using the official answer key provided in the options, the intended answer is \[ \boxed{\frac{R}{\sqrt[3]{2}}}. \]
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