A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the center of the sphere and C is its topmost point. Then

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$|{\vec{V}}_{C} - {\vec{V}}_{A}| = 2 |{\vec{V}}_{B} - {\vec{V}}_{C}|$

${\vec{V}}_{C} - {\vec{V}}_{A} = 2 ({\vec{V}}_{B} - {\vec{V}}_{C})$

${\vec{V}}_{C} - {\vec{V}}_{B} = {\vec{V}}_{B} - {\vec{V}}_{A}$

$|{\vec{V}}_{C} - {\vec{V}}_{A}| = 4 |{\vec{V}}_{B}|$

answer is B, C.

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Detailed Solution

Assuming positive x-direction along the moving direction

$V_{A} = 0, V_{B} = ωr, V_{C} = 2 ωr$ all are along the moving direction of the sphere

Here $\vec{V_{C}} - \vec{V_{B}} = \vec{V_{B}} - \vec{V_{A}}$

$\vec{V_{C}} - \vec{V_{A}}$ is towards the right and $\vec{V_{B}} - \vec{V_{C}}$ is towards left.

Hence, ${\vec{V}}_{C} - {\vec{V}}_{A} = 2 ({\vec{V}}_{B} - {\vec{V}}_{C})$ is incorrect

But considering the magnitude, $|{\vec{V}}_{C} - {\vec{V}}_{A}| = 2 |{\vec{V}}_{B} - {\vec{V}}_{C}|$ is correct

$V_{C} - V_{A} = 2 ωr = 2 V_{B}$

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