Question

A point moves with uniform acceleration and ${{\text{v}}_{\text{1}}}{\text{,}}{{\text{v}}_{\text{2}}}{\text{and }}{{\text{v}}_3}$ denote the average velocities in the three successive intervals of time ${{\text{t}}_{\text{1}}}{\text{,}}{{\text{t}}_{\text{2}}}{\text{and }}{{\text{t}}_{\text{3}}}$. Which of the following relations is correct?
A. $\left( {{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{v}}_{\text{2}}}{\text{ - }}{{\text{v}}_{\text{3}}}} \right){\text{ = }}\left( {{{\text{t}}_{\text{1}}}{\text{ - }}{{\text{t}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{t}}_{\text{2}}}{\text{ + }}{{\text{t}}_{\text{3}}}} \right)$
B. $\left( {{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{v}}_{\text{2}}}{\text{ - }}{{\text{v}}_{\text{3}}}} \right){\text{ = }}\left( {{{\text{t}}_{\text{1}}}{\text{ + }}{{\text{t}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{t}}_{\text{2}}}{\text{ + }}{{\text{t}}_{\text{3}}}} \right)$
C. $\left( {{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{v}}_{\text{2}}}{\text{ - }}{{\text{v}}_{\text{3}}}} \right){\text{ = }}\left( {{{\text{t}}_{\text{1}}}{\text{ - }}{{\text{t}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{t}}_{\text{2}}}{\text{ - }}{{\text{t}}_{\text{3}}}} \right)$
D. $\left( {{{\text{v}}_{\text{1}}}{\text{ - }}{{\text{v}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{v}}_{\text{2}}}{\text{ - }}{{\text{v}}_{\text{3}}}} \right){\text{ = }}\left( {{{\text{t}}_{\text{1}}}{\text{ - }}{{\text{t}}_{\text{2}}}} \right){\text{:}}\left( {{{\text{t}}_{\text{2}}}{\text{ - }}{{\text{t}}_{\text{3}}}} \right)$

Answer 1

Hint: Study of the kinematics deals with the motion of objects. It plays an important role in classical mechanics. It describes the force that causes them to move.

Complete step by step answer:
Given,
Let us consider a point moving in an uniform acceleration and ${{\text{v}}_{\text{1}}}{\text{,}}{{\text{v}}_{\text{2}}}{\text{and }}{{\text{v}}_3}$ denote the average velocities in the three successive intervals of time ${{\text{t}}_{\text{1}}}{\text{,}}{{\text{t}}_{\text{2}}}{\text{and }}{{\text{t}}_{\text{3}}}$.
let ‘u’ be the initial velocity. Then we have the velocity after time $t_1$: ${{\text{v}}_{\text{1}}}{\text{ = u + a}}{{\text{t}}_{\text{1}}}$
velocity after time $\left( {{t_1} + {t_2}} \right):$ ${v_2} = u + a\left( {{t_1} + {t_2}} \right)$
velocity after time $\left( {{t_1} + {t_2} + {t_3}} \right)$: ${v_3} = u + a\left( {{t_1} + {t_2} + {t_3}} \right)$
now, let us consider the average velocity, ${v_1} = \dfrac{{u + {v_1}}}{2} = \dfrac{{u + \left( {u + a{t_1}} \right)}}{2} = u + \dfrac{1}{2}a{t_1}$
${{\text{v}}_{\text{2}}}{\text{ = }}\dfrac{{{{\text{v}}_{\text{1}}}{\text{ + }}{{\text{v}}_{\text{2}}}}}{{\text{2}}}{\text{ = u + a}}{{\text{t}}_{\text{1}}}{\text{ + }}\dfrac{{\text{1}}}{{\text{2}}}{\text{a}}{{\text{t}}_{\text{2}}}$
${{\text{v}}_{\text{2}}}{\text{ = }}\dfrac{{{{\text{v}}_2}{\text{ + }}{{\text{v}}_3}}}{{\text{2}}}{\text{ = u + a}}{{\text{t}}_{\text{1}}}{\text{ + a}}{{\text{t}}_2}{\text{ + }}\dfrac{{\text{1}}}{{\text{2}}}{\text{a}}{{\text{t}}_3}$
After finding the values of $v_1$, $v_2$ and $v_3$
So $\left( {{v_1} - {v_2}} \right) = - \dfrac{1}{2}a\left( {{t_1} + {t_2}} \right)$ …………. (1)
$\left( {{v_2} - {v_3}} \right) = - \dfrac{1}{2}a\left( {{t_2} + {t_3}} \right)$ ……………. (2)
We need to compare the given two equations.
From (1) and (2),
$\therefore \left( {{v_1} - {v_2}} \right):\left( {{v_2} - {v_3}} \right) = \left( {{t_1} + {t_2}} \right):\left( {{t_2} + {t_3}} \right)$
Hence, the correct option is(B).

Additional information:
If an object moving along the straight line covers equal displacement in equal intervals of time, that is, an object moving with uniform velocity is said to be in uniform motion along a straight line.
The displacement of a body in unit time is called its velocity.
If a body has equal displacements in equal intervals of time however small intervals maybe then it is said to be moving with uniform velocity.
For a particle in motion (uniform or non-uniform), the ratio of total displacement to the total time interval is called average velocity.
If the average acceleration over any time interval equals the instantaneous acceleration at any instant of time, then the acceleration is said to be uniform or constant. This does not vary with time. The velocity either increases or decreases at the same rate throughout the motion (or)
If a body has an equal change in velocities in an equal interval of time, however small interval maybe, then it is set to move its uniform acceleration.

Note: The first equation of motion deals with velocity, time, and acceleration. The second equation of motions deals with displacement, velocity, acceleration, and time. The third equation deals with velocity, displacement, and acceleration.