Question

The acceleration of a particle as seen from two frames $S_1$ and $S_2$ has equal magnitude $5m{s}^{-2}$.

• A. The frames must be at rest with respect to each other.
• B. The frames may be moving with respect to each other but neither should be accelerated with respect to the other.
• C. The acceleration of frame $S_2$ with respect to $S_1$ be $0$ or $10m{s}^{-2}$
• D. The acceleration of frame $S_2$ with respect to $S_1$ lies between $0$ and $10m{s}^{-2}$

Answer 1

The correct option is D The acceleration of frame $S_2$ with respect to $S_1$ lies between $0$ and $10m{s}^{-2}$

Using parallelogram law,
Let,
$\overrightarrow{S_1}\overrightarrow{S_2}$ be the arms of the parallelogram,
Given,
$\overrightarrow{S_1}=\overrightarrow{S_2}$
the angle between these two vectors may lie between $0^o$ to ${180}^o$
Let us find the resultant taking $\theta =0^{o}and{180}^o$
$R=\sqrt{{S_1}^2+{S_2}^2-2.S_1{S}_{2}cos\theta }$
$\therefore R=\sqrt{5^2+5^2-2.\left(5\right)\left(5\right)cos{0}^o}=0$
AND
$R=\sqrt{5^2+5^2-2.\left(5\right)\left(5\right)cos{180}^o}=10$
$\therefore$ The acceleration of frame $S_2$ with respect to $S_1$ lies between $0$ and $10m{s}^{-2}$