The position of a particle is given by where is in seconds and in metre. Find the magnitude and direction of velocity at , with respect to the -axis. [2023]

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Q.
The position of a particle is given by $\vec{r} (t) = 4 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}$

where $t$ is in seconds and $r$ in metre. Find the magnitude and direction of velocity $v (t),$ at $t = 1 s$ , with respect to the $x$ -axis. [2023]

1 $4 \sqrt{2} m s^{- 1}, 45 °$

2 $4 \sqrt{2} m s^{- 1}, 60 °$

3 $3 \sqrt{2} m s^{- 1}, 30 °$

4 $3 \sqrt{2} {ms}^{- 1}, 45 °$

Ans.
(1)

$\vec{v} = \frac{d \vec{r}}{d t} = 4 \hat{i} + 4 t \hat{j} + 0 \hat{k}$

At $t = 1 sec;$ $\vec{v} = 4 \hat{i} + 4 (1) \hat{j}$

$Magnitude, | \vec{v} | = \sqrt{{(4)}^{2} + {(4)}^{2}} = 4 \sqrt{2} m/s$

$Direction, \tan θ = \frac{v_{y}}{v_{x}} = 1 or θ = 45 °$

Solution

Q.
The position of a particle is given by $\vec{r} (t) = 4 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}$

where $t$ is in seconds and $r$ in metre. Find the magnitude and direction of velocity $v (t),$ at $t = 1 s$ , with respect to the $x$ -axis. [2023]

1 $4 \sqrt{2} m s^{- 1}, 45 °$

2 $4 \sqrt{2} m s^{- 1}, 60 °$

3 $3 \sqrt{2} m s^{- 1}, 30 °$

4 $3 \sqrt{2} {ms}^{- 1}, 45 °$

Ans.
(1)

$\vec{v} = \frac{d \vec{r}}{d t} = 4 \hat{i} + 4 t \hat{j} + 0 \hat{k}$

At $t = 1 sec;$ $\vec{v} = 4 \hat{i} + 4 (1) \hat{j}$

$Magnitude, | \vec{v} | = \sqrt{{(4)}^{2} + {(4)}^{2}} = 4 \sqrt{2} m/s$

$Direction, \tan θ = \frac{v_{y}}{v_{x}} = 1 or θ = 45 °$

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Solution

Q. The position of a particle is given by r→(t)=4t i^+2t2 j^+5 k^ where t is in seconds and r in metre. Find the magnitude and direction of velocity v(t), at t=1s, with respect to the x-axis. [2023]

1 42 ms-1, 45°

2 42 ms-1, 60°

3 32 ms-1, 30°

4 32 ms-1, 45°

Ans. (1) v→=dr→dt=4i^+4tj^+0k^ At t=1 sec; v→=4i^+4(1)j^ Magnitude, |v→|=(4)2+(4)2=42 m/s Direction, tanθ=vyvx=1 or θ=45°

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Q.
The position of a particle is given by $\vec{r} (t) = 4 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}$

where $t$ is in seconds and $r$ in metre. Find the magnitude and direction of velocity $v (t),$ at $t = 1 s$ , with respect to the $x$ -axis. [2023]

1 $4 \sqrt{2} m s^{- 1}, 45 °$

2 $4 \sqrt{2} m s^{- 1}, 60 °$

3 $3 \sqrt{2} m s^{- 1}, 30 °$

4 $3 \sqrt{2} {ms}^{- 1}, 45 °$

Ans.
(1)

$\vec{v} = \frac{d \vec{r}}{d t} = 4 \hat{i} + 4 t \hat{j} + 0 \hat{k}$

At $t = 1 sec;$ $\vec{v} = 4 \hat{i} + 4 (1) \hat{j}$

$Magnitude, | \vec{v} | = \sqrt{{(4)}^{2} + {(4)}^{2}} = 4 \sqrt{2} m/s$

$Direction, \tan θ = \frac{v_{y}}{v_{x}} = 1 or θ = 45 °$