Question

The accelerations of a particle as seen from two frames ${{S}_{1}}$ and ${{S}_{2}}$ have 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 ${{S}_{2}}$ with respect to ${{S}_{1}}$ may either be zero or$10m{{s}^{-2}}$
(d) The acceleration of ${{S}_{2}}$ with respect to ${{S}_{1}}$ may be anything between zero and $10m{{s}^{-2}}$.

Answer 1

The rate of change of an object's velocity with respect to time is called acceleration in mechanics. Accelerations are values that are measured in vectors (in that they have magnitude and direction). The orientation of the net force applied on an item determines the orientation of its acceleration. Newton's Second Law describes the magnitude of an object's acceleration. We use this concept here.

Complete step by step solution:
The symbol alpha ($\alpha$) is used to indicate angular acceleration, which is measured in units of angle per unit time squared (radians per second squared in SI units). Angular acceleration is a pseudoscalar in two dimensions, with a positive sign when the angular speed rises counterclockwise or decreases clockwise, and a negative sign when the angular speed increases clockwise or decreases counterclockwise. Angular acceleration is a pseudovector in three dimensions.

The simplest form of the parallelogram rule (also known as the parallelogram identity) is found in fundamental geometry in mathematics. The total of the squares of the lengths of a parallelogram's four sides equals the sum of the squares of the lengths of the two diagonals, according to this rule. The sides are denoted by the letters AB, BC, CD, and DA. However, because a parallelogram in Euclidean geometry must have equal opposing sides, AB = CD and BC = DA, the law may be written as $2A{{B}^{2}}+2B{{C}^{2}}=A{{C}^{2}}+B{{D}^{2}}$
Using parallelogram law, Let,
$\overrightarrow{S_{1}} \vec{S}_{2}$ be the arms of the parallelogram, Given, $\overrightarrow{\mathbf{S}_{1}}=\overrightarrow{\mathbf{S}_{2}}$
the angle between these two vectors lie between $0^{\circ}$ to $180^{\circ}$
so let us find resultant by taking $\theta=0^{\circ}$ and $180^{\circ}$
$\mathrm{R}=\sqrt{\mathbf{S}_{1}^{2}+\mathbf{S}_{2}{ }^{2}-\mathbf{2} \cdot \mathbf{S}_{1} \mathbf{S}_{2} \cos \theta}$
$\therefore \mathrm{R}=\sqrt{5^{2}+5^{2}-2 \cdot(5)(5) \cos 0^{\circ}}=0$
AND $\mathrm{R}=\sqrt{5^{2}+5^{2}-2 \cdot(5)(5) \cos 180^{\circ}}=10$
$\therefore$ The acceleration of frame $\mathrm{S}_{2}$ with respect to $\mathrm{S}_{1}$ lies between 0 and $10 \mathrm{~ms}^{-2}$
Hence option D is correct.

Note:
A frame of reference (or reference frame) in physics and astronomy is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points, geometric points whose position is identified both mathematically (with numerical coordinate values) and physically (with physical coordinate values) (signaled by conventional markers).