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Motion in a Plane — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane1 Mark
MCQ
Following are four differrent relations about displacement, velocity and acceleration for the motion of a particle in general. Choose the incorrect one $(s):$
  • A
    $\text{v}_\text{av}=\frac{1}2\big[\text{v}(\text{t}_1)+\text{v}(\text{t}_2)\big]$
  • B
    $\text{v}_\text{av}=\frac{r(\text{t}_2)-\text{r}(\text{t}_1)}{\text{t}_2-\text{t}_1}$
  • C
    $\text{r}=\frac{1}2(\text{v}(\text{t}_2)-\text{v}(\text{t}_1))(\text{t}_2-\text{t}_1)$
  • ✓
    Both $A$ and $C$
✓

Answer

Correct option: D.
Both $A$ and $C$
When an object covers a displacement $\Delta\text{r}$ in time $\Delta\text{t},$ its average velocity is given by $\vec{\text{v}}_\text{avg}=\frac{\overrightarrow{\Delta\text{r}}}{\Delta\text{t}}=\frac{\text{r}_2-\text{r}_1}{\text{t}_2-\text{t}_1}$ where $r_1$ and $r_2$ are position vectors corresponding to time $t_1$ and $t_2$.
If the velocity of an object changes from $v_1$ to $v_2$ in time $\Delta\text{t},$ average acceleration is given by
$\text{a}_\text{av}=\frac{\Delta\text{v}}{\Delta\text{t}}=\frac{\text{v}_2-\text{v}_1}{\text{t}_2-\text{t}_1}$
But, when acceleration is non$-$uniform,
$\text{v}_\text{av}\neq\frac{\text{v}_1+\text{v}_2}{2}$
Option $(c)$ is similar to the relation $\vec{\text{r}}=\frac{1}2\text{at}^2$ which is not correct if initial velocity is given.
So $(b)$ and $(c)$ are the correct relations for the uniform acceleration.

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