Motion In A Plane - Important Concept For NEET

Q 1 :
The two dimensional motion of a particle, described by $\vec{r} = (\hat{i} + 2 \hat{j}) A \cos ω t$ is a/an

A. parabolic path
B. elliptical path
C. periodic path
D. simple harmonic motion

Choose the correct answer from the options given below: [2024]

B, C and D only
A, B and C only
A, C and D only
C and D only

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(4)

$\vec{r} = A \cos ω t \hat{i} + 2 A \cos ω t \hat{j}$

$x = A \cos ω t$ ...(i)

$y = 2 A \cos ω t$ ...(ii)

Dividing (i) by (ii), we get

$\frac{x}{y} = \frac{1}{2} [Equation of straight line]$

Q 2 :
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]

$4 \sqrt{2} m s^{- 1}, 45 °$
$4 \sqrt{2} m s^{- 1}, 60 °$
$3 \sqrt{2} m s^{- 1}, 30 °$
$3 \sqrt{2} {ms}^{- 1}, 45 °$

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(1)

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

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

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

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

Q 3 :
The $x$ and $y$ coordinates of the particle at any time are $x = 5 t - 2 t^{2}$ and $y = 10 t$ respectively, where $x$ and $y$ are in metres and $t$ in seconds. The acceleration of the particle at $t = 2 s$ is [2017]

$5 {ms}^{- 2}$
$- 4 {ms}^{- 2}$
$- 8 {ms}^{- 2}$
$0$

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(2)

$x = 5 t - 2 t^{2}, y = 10 t$

$\frac{d x}{d t} = 5 - 4 t, \frac{d y}{d t} = 10 ∴$ $v_{x} = 5 - 4 t, v_{y} = 10$

$\frac{d v_{x}}{d t} = - 4, \frac{d v_{y}}{d t} = 0 ∴$ $a_{x} = - 4, a_{y} = 0$

$Acceleration, \vec{a} = a_{x} \hat{i} + a_{y} \hat{j} = - 4 \hat{i}$

$∴$ $The acceleration of the particle at t = 2 s is - 4 {ms}^{- 2} .$

Q 4 :
The position vector of a particle $\vec{R}$ as a function of time is given by $\vec{R} = 4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}$ where R is in meters, t is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along x and y-directions, respectively. Which one of the following statements is wrong for the motion of the particle? [2015]

Magnitude of the velocity of particle is 8 meter/second.
Path of the particle is a circle of radius 4 meter.
Acceleration vector is along $- \vec{R}$ .
Magnitude of acceleration vector is $\frac{v^{2}}{R}$ , where $v$ is the velocity of particle.

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(1)

Here, $\vec{R} = 4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}$

The velocity of the particle is

$\vec{v} = \frac{d \vec{R}}{d t} = \frac{d}{d t} [4 \sin (2 π t) \hat{i} + 4 \cos (2 π t) \hat{j}] = 8 π \cos (2 π t) \hat{i} - 8 π \sin (2 π t) \hat{j}$

Its magnitude is $| \vec{v} |$ $= \sqrt{(8 π \cos {(2 π t))}^{2} + (- 8 π \sin {(2 π t))}^{2}}$

$= \sqrt{64 π^{2} \cos^{2} (2 π t) + 64 π^{2} \sin^{2} (2 π t)}$

$= \sqrt{64 π^{2} [\cos^{2} (2 π t) + \sin^{2} (2 π t)]}$

$= \sqrt{64 π^{2}} (As \sin^{2} θ + \cos^{2} θ = 1)$

$= 8 π m/s$

Q 5 :
A particle is moving such that its position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m) at time t = 2 s and (13 m, 14 m) at time t = 5 s. Average velocity vector $({\vec{v}}_{av})$ from t = 0 to t = 5 s is [2014]

$\frac{1}{5} (13 \hat{i} + 14 \hat{j})$
$\frac{7}{3} (\hat{i} + \hat{j})$
$2 (\hat{i} + \hat{j})$
$\frac{11}{5} (\hat{i} + \hat{j})$

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(4)

At time t = 0, the position vector of the particle is ${\vec{r}}_{1} = 2 \hat{i} + 3 \hat{j} .$

At time t = 5 s, the position vector of the particle is ${\vec{r}}_{2} = 13 \hat{i} + 14 \hat{j} .$

Displacement from ${\vec{r}}_{1}$ to ${\vec{r}}_{2}$ is

$Δ \vec{r} = {\vec{r}}_{2} - {\vec{r}}_{1} = (13 \hat{i} + 14 \hat{j}) - (2 \hat{i} + 3 \hat{j}) = 11 \hat{i} + 11 \hat{j}$

$∴ Average velocity, {\vec{v}}_{av} = \frac{Δ \vec{r}}{Δ t} = \frac{11 \hat{i} + 11 \hat{j}}{5 - 0} = \frac{11}{5} (\hat{i} + \hat{j})$

Topic Question Set

Q 1 :
The two dimensional motion of a particle, described by $\vec{r} = (\hat{i} + 2 \hat{j}) A \cos ω t$ is a/an

A. parabolic path
B. elliptical path
C. periodic path
D. simple harmonic motion

Choose the correct answer from the options given below: [2024]

Q 2 :
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]

Q 3 :
The $x$ and $y$ coordinates of the particle at any time are $x = 5 t - 2 t^{2}$ and $y = 10 t$ respectively, where $x$ and $y$ are in metres and $t$ in seconds. The acceleration of the particle at $t = 2 s$ is [2017]

Q 5 :
A particle is moving such that its position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m) at time t = 2 s and (13 m, 14 m) at time t = 5 s. Average velocity vector $({\vec{v}}_{av})$ from t = 0 to t = 5 s is [2014]

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Topic Question Set

Q 1 : The two dimensional motion of a particle, described by r→=(i^+2j^)Acosωt is a/an A. parabolic path B. elliptical path C. periodic path D. simple harmonic motion Choose the correct answer from the options given below: [2024]

Q 2 : 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]

Q 3 : The x and y coordinates of the particle at any time are x=5t-2t2 and y=10t respectively, where x and y are in metres and t in seconds. The acceleration of the particle at t=2 s is [2017]

Q 5 : A particle is moving such that its position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m) at time t = 2 s and (13 m, 14 m) at time t = 5 s. Average velocity vector (v→av)from t = 0 to t = 5 s is [2014]

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Q 1 :
The two dimensional motion of a particle, described by $\vec{r} = (\hat{i} + 2 \hat{j}) A \cos ω t$ is a/an

A. parabolic path
B. elliptical path
C. periodic path
D. simple harmonic motion

Choose the correct answer from the options given below: [2024]

Q 2 :
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]

Q 3 :
The $x$ and $y$ coordinates of the particle at any time are $x = 5 t - 2 t^{2}$ and $y = 10 t$ respectively, where $x$ and $y$ are in metres and $t$ in seconds. The acceleration of the particle at $t = 2 s$ is [2017]

Q 5 :
A particle is moving such that its position coordinates (x, y) are (2 m, 3 m) at time t = 0, (6 m, 7 m) at time t = 2 s and (13 m, 14 m) at time t = 5 s. Average velocity vector $({\vec{v}}_{av})$ from t = 0 to t = 5 s is [2014]