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Motion in a Plane — Physics STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane1 Mark
Question
Following are four differrent relations about displacement, velocity and acceleration for the motion of a particle in general. Choose the incorrect one (s):

  1. $\text{v}_\text{av}=\frac{1}2\big[\text{v}(\text{t}_1)+\text{v}(\text{t}_2)\big]$

  2. $\text{v}_\text{av}=\frac{r(\text{t}_2)-\text{r}(\text{t}_1)}{\text{t}_2-\text{t}_1}$

  3. $\text{r}=\frac{1}2(\text{v}(\text{t}_2)-\text{v}(\text{t}_1))(\text{t}_2-\text{t}_1)$

  4. $\text{a}_\text{av}=\frac{\text{v}(\text{t}_2)-\text{v}(\text{t}_1)}{\text{t}_2-\text{t}_1}$

✓

Answer

  1. $\text{v}_\text{av}=\frac{1}2\big[\text{v}(\text{t}_1)+\text{v}(\text{t}_2)\big]$
  1. $\text{r}=\frac{1}2(\text{v}(\text{t}_2)-\text{v}(\text{t}_1))(\text{t}_2-\text{t}_1)$

Explanation:

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 r1 and r2 are position vectors corresponding to time t1 and t2.

If the velocity of an object changes from v1 to v2 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 (d) are the correct relations for the uniform acceleration.

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