Question

The acceleration of a particle is seen from two frames $S_1$​ and $S_2$ have an equal magnitude of $5m{s}^{-2}$. then

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

Answer 1

The correct option is B

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

Step 1. Given data

1. $\overrightarrow{S_1}$ and $\overrightarrow{S_2}$ is of equal magnitude
2. $\left|\overrightarrow{S_1}\right|=\left|\overrightarrow{S_2}\right|=5m/s^2$

Step 2. Formula used

Consider a vector $\overrightarrow{v}$ represented by $AB$ side of the parallelogram and the vector $\overrightarrow{u}$ represented by $AD$ of the parallelogram $ABCD$ then the resultant vector $\overrightarrow{w}$ represented by $AC$ is

$\overrightarrow{w}=\sqrt{{\left|\overrightarrow{u}\right|}^2+{\left|\overrightarrow{v}\right|}^2-2·\left|\overrightarrow{u}\right|·\left|\overrightarrow{v}\right|·\cos \left(\phi \right)}$

Step 3. Find the acceleration of frame $S_2$​ with respect to $S_1$

Using parallelogram law,

Let $\overrightarrow{S_1}$​ and $\overrightarrow{S_2}$ are the two arms of the parallelogram

Given $\left|\overrightarrow{S_1}\right|=\left|\overrightarrow{S_2}\right|=5m/s^2$

The angle between the two vectors may lie between $0°$ to $180°$

The resultant when the angle is $0°$ is

$$\begin{gathered} R=\sqrt{S_1^2+S_2^2-2·S_1·S_2·\cos \left(\theta \right)} \\ =\sqrt{5^2+5^2-2·5·5\cos \left(0°\right)} \\ =\sqrt{50-50} \\ =0 \end{gathered}$$

The resultant when the angle is $180°$ is

$$\begin{gathered} R=\sqrt{S_1^2+S_2^2-2·S_1·S_2·\cos \left(\theta \right)} \\ =\sqrt{5^2+5^2-2·5·5\cos \left(180°\right)} \\ =\sqrt{50+50} \\ =10 \end{gathered}$$

Therefore the acceleration of the frame $S_2$​ with respect to $S_1$​ lies between $0$ and $10m{s}^{-2}$.

Hence, Option B is correct