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

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

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

Q 6 :

The $(x, y, z)$ coordinates of two points $A$ and $B$ are given respectively as (0, 4, - 2) and (- 2, 8, - 4). The displacement vector from $A$ to $B$ is

$- 2 \hat{i} + 4 \hat{j} - 2 \hat{k}$
$2 \hat{i} - 4 \hat{j} + 2 \hat{k}$
$2 \hat{i} + 4 \hat{j} - 2 \hat{k}$
$- 2 \hat{i} - 4 \hat{j} - 2 \hat{k}$

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(1)

$Here, {\vec{r}}_{A} = 0 \hat{i} + 4 \hat{j} - 2 \hat{k}, {\vec{r}}_{B} = - 2 \hat{i} + 8 \hat{j} - 4 \hat{k}$

$Displacement vector from A to B is given by$

$\vec{r} = {\vec{r}}_{B} - {\vec{r}}_{A} = (- 2 \hat{i} + 8 \hat{j} - 4 \hat{k}) - (0 \hat{i} + 4 \hat{j} - 2 \hat{k}) = - 2 \hat{i} + 4 \hat{j} - 2 \hat{k}$

Q 7 :

A person moves 30 m north, then 30 m east, then $30 \sqrt{2}$ m south-west. His displacement from the original position is

zero
28 m towards south
10 m towards west
15 m towards east

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(1)

Resolving displacement $30 \sqrt{2}$ m south-west into two rectangular components, we get

Displacement in south $= 30 \sqrt{2} \cos 45 ° = 30 \sqrt{2} \times (\frac{1}{\sqrt{2}}) = 30 m$

Displacement in west $= 30 \sqrt{2} \sin 45 ° = 30 \sqrt{2} \times (\frac{1}{\sqrt{2}}) = 30 m$

$∴$ Effective displacement due to 30 m north and 30 m south is zero.

Also effective displacement due to 30 m east and 30 m west is zero.

Q 8 :

A bird flies from (-3 m, 4 m, -3 m) to (7 m, -2 m, -3 m) in the xyz-coordinates. The bird's displacement vector is given by

$(4 \hat{i} + 2 \hat{j} - 6 \hat{k})$
$(10 \hat{i} - 6 \hat{j})$
$(4 \hat{i} - 2 \hat{j})$
$(10 \hat{i} + 6 \hat{j} - 6 \hat{k})$

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(2)

$Bird's displacement is Δ \vec{r} = {\vec{r}}_{2} - {\vec{r}}_{1}$

$∴$ $Δ \vec{r} = (7 \hat{i} - 2 \hat{j} - 3 \hat{k}) - (- 3 \hat{i} + 4 \hat{j} - 3 \hat{k}) = 10 \hat{i} - 6 \hat{j}$

Q 9 :

A particle starts moving from point (2, 10, 1). Displacement for the particle is $8 \hat{i} - 2 \hat{j} + \hat{k}$ . The final coordinates of the particle is

(10, 8, 2)
(8, 10, 2)
(2, 10, 8)
(8, 2, 10)

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(1)

$Initial position vector of the particle, {\vec{r}}_{i} = 2 \hat{i} + 10 \hat{j} + \hat{k}$

$Let final position vector of the particle be$

${\vec{r}}_{f} = x \hat{i} + y \hat{j} + z \hat{k}$

$∴$ $Displacement, Δ \vec{r} = {\vec{r}}_{f} - {\vec{r}}_{i}$

$8 \hat{i} - 2 \hat{j} + \hat{k} = x \hat{i} + y \hat{j} + z \hat{k} - (2 \hat{i} + 10 \hat{j} + \hat{k})$

$or x \hat{i} + y \hat{j} + z \hat{k} = 10 \hat{i} + 8 \hat{j} + 2 \hat{k}$

$Hence, the final coordinates of the particle are (10, 8, 2) .$

Q 10 :

For any arbitrary motion in space, which of the following relations is true?

(The average stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

${\vec{v}}_{average} = \frac{1}{2} [\vec{v} (t_{1}) + \vec{v} (t_{2})]$
${\vec{v}}_{average} = \frac{\vec{r} (t_{2}) - \vec{r} (t_{1})}{t_{2} - t_{1}}$
$\vec{v} (t) = \vec{v} (0) + \vec{a} t$
$\vec{r} (t) = \vec{r} (0) + \vec{v} (0) t + \frac{1}{2} \vec{a} t^{2}$

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(2)

The relation (2) is true, others are false because relations (1), (3) and (4) hold only for uniformly accelerated motion.

Q 11 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The direction of velocity of the particle at t = 1 s is

53° with $x$ -axis
37° with $x$ -axis
30° with $y$ -axis
60° with $y$ -axis

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(1)

$Given : \vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}$

$Velocity, \vec{v} = \frac{d \vec{r}}{d t} = \frac{d}{d t} (3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}) = 3 \hat{i} + 4 t \hat{j} {m s}^{- 1}$

$Let θ be the angle which the direction of \vec{v} makes with the x -axis. Then$

$\tan θ = \frac{v_{y}}{v_{x}} = \frac{4 t}{3} = \frac{4}{3}$ $or$ $θ = \tan^{- 1} (\frac{4}{3}) = 53 °$

Q 12 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The acceleration of the particle at t = 1 s is

$2 \hat{j} {m s}^{- 2}$
$- 2 \hat{j} {m s}^{- 2}$
$4 \hat{j} {m s}^{- 2}$
$- 4 \hat{j} {m s}^{- 2}$

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(3)

$Given : \vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}$

$Velocity, \vec{v} = \frac{d \vec{r}}{d t} = \frac{d}{d t} (3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k}) = 3 \hat{i} + 4 t \hat{j} {m s}^{- 1}$

$\vec{v} = 3 \hat{i} + 4 t \hat{j} {m s}^{- 1}$

$∴$ $Acceleration, \vec{a} = \frac{d \vec{v}}{d t} = 4 \hat{j} {m s}^{- 2}$

$Acceleration of the particle remains constant all the time.$

Q 13 :

If $x = 5 t + 3 t^{2}$ and $y = 4 t$ are the $x$ and $y$ coordinates of a particle at any time $t$ second where $x$ and $y$ are in metre, then the acceleration of the particle

is zero throughout its motion
is a constant throughout its motion
depends only on its $y$ component
varies along both $x$ and $y$ direction

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(2)

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]

Q 6 :

The $(x, y, z)$ coordinates of two points $A$ and $B$ are given respectively as (0, 4, - 2) and (- 2, 8, - 4). The displacement vector from $A$ to $B$ is

Q 7 :

A person moves 30 m north, then 30 m east, then $30 \sqrt{2}$ m south-west. His displacement from the original position is

Q 8 :

A bird flies from (-3 m, 4 m, -3 m) to (7 m, -2 m, -3 m) in the xyz-coordinates. The bird's displacement vector is given by

Q 9 :

A particle starts moving from point (2, 10, 1). Displacement for the particle is $8 \hat{i} - 2 \hat{j} + \hat{k}$ . The final coordinates of the particle is

Q 10 :

For any arbitrary motion in space, which of the following relations is true?

(The average stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

Q 11 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The direction of velocity of the particle at t = 1 s is

Q 12 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The acceleration of the particle at t = 1 s is

Q 13 :

If $x = 5 t + 3 t^{2}$ and $y = 4 t$ are the $x$ and $y$ coordinates of a particle at any time $t$ second where $x$ and $y$ are in metre, then the acceleration of the particle

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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]

Q 6 : The (x, y, z) coordinates of two points A and B are given respectively as (0, 4, - 2) and (- 2, 8, - 4). The displacement vector from A to B is

Q 7 : A person moves 30 m north, then 30 m east, then 302 m south-west. His displacement from the original position is

Q 8 : A bird flies from (-3 m, 4 m, -3 m) to (7 m, -2 m, -3 m) in the xyz-coordinates. The bird's displacement vector is given by

Q 9 : A particle starts moving from point (2, 10, 1). Displacement for the particle is 8i^−2j^+k^. The final coordinates of the particle is

Q 10 : For any arbitrary motion in space, which of the following relations is true? (The average stands for average of the quantity over the time interval t1 to t2)

Q 11 : The position of a particle is given by r→=3ti^+2t2j^+5k^, where t is in seconds and the coefficients have the proper units for r→ to be in metres. The direction of velocity of the particle at t = 1 s is

Q 12 : The position of a particle is given by r→=3ti^+2t2j^+5k^, where t is in seconds and the coefficients have the proper units for r→ to be in metres. The acceleration of the particle at t = 1 s is

Q 13 : If x=5t+3t2 and y=4t are the x and y coordinates of a particle at any time t second where x and y are in metre, then the acceleration of the particle

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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]

Q 6 :

The $(x, y, z)$ coordinates of two points $A$ and $B$ are given respectively as (0, 4, - 2) and (- 2, 8, - 4). The displacement vector from $A$ to $B$ is

Q 7 :

A person moves 30 m north, then 30 m east, then $30 \sqrt{2}$ m south-west. His displacement from the original position is

Q 8 :

A bird flies from (-3 m, 4 m, -3 m) to (7 m, -2 m, -3 m) in the xyz-coordinates. The bird's displacement vector is given by

Q 9 :

A particle starts moving from point (2, 10, 1). Displacement for the particle is $8 \hat{i} - 2 \hat{j} + \hat{k}$ . The final coordinates of the particle is

Q 10 :

For any arbitrary motion in space, which of the following relations is true?

(The average stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

Q 11 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The direction of velocity of the particle at t = 1 s is

Q 12 :

The position of a particle is given by $\vec{r} = 3 t \hat{i} + 2 t^{2} \hat{j} + 5 \hat{k},$ where $t$ is in seconds and the coefficients have the proper units for $\vec{r}$ to be in metres. The acceleration of the particle at t = 1 s is

Q 13 :

If $x = 5 t + 3 t^{2}$ and $y = 4 t$ are the $x$ and $y$ coordinates of a particle at any time $t$ second where $x$ and $y$ are in metre, then the acceleration of the particle