Question

A particle moves along positive branch of the curve, $y=\frac{x}{2}$, where $x=\frac{t^3}{3}$, $x$ and $y$ are measured in maters and $t$ in second, then:

• A. The velocity of particle at $t=1s$ is $\hat{i}+\frac{1}{2}\hat{j}$
• B. The velocity of particle at $t=1s$ is $\frac{1}{2}\hat{i}+\hat{j}$
• C. The acceleration of particle at $t=2s$ is $2\hat{i}+\hat{j}$
• D. The acceleration of particle at $t=2s$ is $\hat{i}+2\hat{j}$

Answer 1

Answer: (A), (C)

The correct options are
A The velocity of particle at $t=1s$ is $\hat{i}+\frac{1}{2}\hat{j}$
C The acceleration of particle at $t=2s$ is $2\hat{i}+\hat{j}$

$\left(A\right)$

$x=\frac{t^3}{3}$

$y=\frac{x}{2}=\frac{t^3}{6}$

$\overrightarrow{r}=x\hat{i}+y\hat{j}$

$\overrightarrow{r}=\frac{t^3}{3}\hat{i}+\frac{t^3}{6}\hat{j}$

$\overrightarrow{V}=\frac{d\overrightarrow{r}}{dt}=t^2\hat{i}+\frac{1}{2}{t}^2\hat{j}$

$\overrightarrow{a}=\frac{d\overrightarrow{V}}{dt}=2t\hat{i}+t\hat{j}$

At $t=1$,

$\overrightarrow{V}=\hat{i}+\frac{1}{2}\hat{j}$

$\overrightarrow{a}=2\hat{i}+\hat{j}$

At $t=2$,

$\overrightarrow{V}=4\hat{i}+2\hat{j}$

$\overrightarrow{a}=4\hat{i}+2\hat{j}$