Question

The radius vector describing the position of the particle $A$ relative to origin is $\overrightarrow{r}=t^2\hat{i}+(t-1{)}^2\hat{j}$.
Find the rectangular components of the average velocity in the time interval between $t$ and $t+\Delta t$ ?

• A. $(\Delta t+2t),(\Delta t+2t-2)$
• B. $(\Delta t-2t),(\Delta t+2t-2)$
• C. $(\Delta t-t),(\Delta t+2t+2)$
• D. $(\Delta t+t),(\Delta t+2t)$

Answer 1

The correct option is A $(\Delta t+2t),(\Delta t+2t-2)$

$$\begin{array}{c}\overrightarrow{r}=t^2\hat{i}+(t-1{)}^2\hat{j}\\ \text{At}t=t\\ \overrightarrow{r_1}=t^2\hat{i}+(t-1{)}^2\hat{j}\\ \text{At}t=t+\Delta t\\ \overrightarrow{r_2}=(t+\Delta t{)}^2\hat{i}+(t+\Delta t-1{)}^2\hat{j}\\ \overrightarrow{r_2}-\overrightarrow{r_1}=\hat{i}\left[\right(t+\Delta t{)}^2-t^2]+\hat{j}[\Delta t^2+2\Delta t(t-1\left)\right]\\ \overrightarrow{r_2}-\overrightarrow{r_1}=\hat{i}[\Delta t^2+2t\Delta t]+\hat{j}[\Delta t^2+2\Delta t(t-1\left)\right]\\ v_{avg}=\frac{\overrightarrow{r_2}-\overrightarrow{r_1}}{\Delta t}\\ v_{avg}=(\Delta t+2t)\hat{i}+(\Delta t+2t-2)\hat{j}\end{array}$$