Displacement in 2D Motion

Q. Two persons of masses $55\text{kg}$ and $65\text{kg}$ respectively, are at the opposite ends of a boat. The length of the boat is $3\text{m}$ and weighs $100\text{kg}$. The $55\text{kg}$ man walks up to the $65\text{kg}$ man and sits with him. If the boat is in still water the centre of mass of the system shifts by

1. $3\text{m}$
2. $2.3\text{m}$
3. $0.75\text{m}$
4. Zero

Q. A rain drop of mass $1\text{g}$ falling from a height of $1\text{km}$ hits the ground with a speed of $50\text{m/s}$. If a resistive force acts on the drop, then the work done by the resistive force is (Take $g:10{\text{m/s}}^2)$

1. $-10\text{J}$
2. $10\text{J}$
3. $8.75\text{J}$
4. $-8.75\text{J}$

Q. Suppose you are told that the linear size of everything in the universe has been doubled overnight can you tell the statement by measuring sizes with a metre stick ? can you test it by using the fact that the speed of light is universal constant and has not changed ? what will happen if all the clocks in the universe also start running at half the speed ?

Q.

In Figure shows a uniform rod of length 30 cm having a mass of 3.0 kg. The strings shown in the figure are pulled by constant forces of 20 N and 32 N. Find the force exerted by the 20 cm part of the rod on the 10 cm part. All the surfaces are smooth and the strings and the pulleys are light.

Q. Two particles A and B are moving in XY plane. Their positions vary with time $t$, according to relation
$x_A\left(t\right)=4t^2,x_B\left(t\right)=7$
$y_A\left(t\right)=3t,y_B\left(t\right)=3+4t^2$
Distance between these two particles at $t=1\text{s}$ is:-

1. $5\text{m}$
2. $3\text{m}$
3. $4\text{m}$
4. $\sqrt{12}\text{m}$

Q.

A cylindrical vessel, whose diameter and height both are equal to 30 cm, is placed on a horizontal surface and a small particle P is placed in it at a distance of 5.0 cm from the centre. An eye is placed at a position such that the edge of the bottom is just visible (see Figure). The particle P is in the plane of drawing. Up to what minimum height should water be poured in the vessel to make the particle P visible ?

Q. An aeroplane moves $400\text{m}$ towards north, $300\text{m}$ towards west and then $1200\text{m}$ vertically upward, then its displacement from the initial position is

1. $1600\text{m}$
2. $1800\text{m}$
3. $1500\text{m}$
4. $1300\text{m}$

Q.

What do you mean by acceleration?

Q. Train $A$ and train $B$ are running on parallel tracks in the opposite directions with speeds of $36{\text{km h}}^{-1}$ and $72{\text{km h}}^{-1}$, respectively. A person is walking in train $A$ in the direction opposite to its motion with a speed of $1.8{\text{km h}}^{-1}$. The speed $\left({\text{in ms}}^{–1}\right)$ of this person as observed from train $B$ will be close to

1. $29.5{\text{ms}}^{-1}$
2. $28.5{\text{ms}}^{-1}$
3. $31.5{\text{ms}}^{-1}$
4. $30.5{\text{ms}}^{-1}$

Q. A body is thrown with a velocity equal to $n$ times the escape velocity $\left(v_e\right)$. Velocity of the body at a large distance away will be

1. $v_e\sqrt{n^2-1}$
   
2. $v_e\sqrt{n^2+1}$
3. $v_e\sqrt{1-n^2}$
   
4. None of these

Q. A system of smooth hollow cylinder and block was initially at rest as shown. If cylinder is given constant acceleration 2g in horizontal direction, then the maximum angular displacement of the block with the vertical is:

1. $2ta{n}^{-1}\left(\frac{1}{2}\right)$
2. $2ta{n}^{-1}\left(2\right)$
3. $ta{n}^{-1}\left(2\right)$
4. $ta{n}^{-1}\left(1\right)$

Q.

The equation of motion of a particle is $\frac{d^{2}y}{d{t}^2}+Ky=0$ , where K is positive

constant. The time period of the motion is given by

1. 

2. 

3. 

4. 

Q.

An athlete completes one round of a circular track of diameter $200m$ in $40s$ What will be the distance covered and the displacement at the end of $2min.20s.$

Q. The graph in figure given below represents the velocity-time graph of a particle, starting from rest at point $\text{P}$. Find the time when particle will reach point $\text{P}$ again.

1. $8\text{s}$
2. $10\text{s}$
3. $12\text{s}$
4. $16\text{s}$

Q.

A smooth square platform ABCD is moving towards right with a uniform speed v. At what angle $\theta$ must a particle be projected from A with speed u so that it strikes the point B?

1. $co{s}^{-1}\left(\frac{v}{u}\right)$

2. $co{s}^{-1}\left(\frac{u}{v}\right)$

3. $si{n}^{-1}\left(\frac{v}{u}\right)$

4. $si{n}^{-1}\left(\frac{u}{v}\right)$

Q. The position vector of a particle is determined by the expression $\overline{r}$ = 3 $t^2$ $\hat{i}$ + 4 $t^2$$\hat{j}$ + 7$\hat{k}$ .The distance traversed in first 10 sec is

1. 300 m
2. 150 m
3. 100 m
4. 500 m

Q. The displacement $x$ of a simple harmonic oscillator varies with time $t$ as
$x\left(t\right)=0.5\sin (2\pi t+\frac{\pi }{4})$
What is the magnitude of maximum acceleration ?

1. $\frac{8}{2}{\pi }^2\frac{\text{m}}{{\text{s}}^2}$
2. $5{\pi }^2\frac{\text{m}}{{\text{s}}^2}$
3. $2{\pi }^2\frac{\text{m}}{{\text{s}}^2}$
4. $4{\pi }^2\frac{\text{m}}{{\text{s}}^2}$

Q. The velocity $\left(v\right)$ and time $\left(t\right)$ graph of a body in a straight line motion is shown in the figure. The point $S$ is at $4.333\text{s}$ seconds. The total distance covered by the body in $6\text{s}$ is:

1. $\frac{37}{3}\text{m}$
2. $12\text{m}$
3. $11\text{m}$
4. $\frac{49}{4}\text{m}$

Q. A carom board $(4ft\times 4ftsquare)$ has the queen at the centre. The queen, hit by the striker moves to the front edge, rebounds and goes in the hole behind the striking line. Find the magnitude of displacement of the queen
$\left(i\right)$ from the centre to the front edge
$\left(ii\right)$ from the front edge to the hole and
$\left(iii\right)$ from the centre of the hole.

1. $\left(i\right)$ $\frac{2}{3}\sqrt{10}ft$ $\left(ii\right)$ $\frac{4}{3}\sqrt{10}ft$ $\left(iii\right)$ $2\sqrt{2}ft$
2. $\left(i\right)$ $\frac{4}{3}\sqrt{10}ft$ $\left(ii\right)$ $\frac{4}{3}\sqrt{10}ft$ $\left(iii\right)$ $2\sqrt{2}ft$
3. $\left(i\right)$ $\frac{4}{3}\sqrt{10}ft$ $\left(ii\right)$ $\frac{2}{3}\sqrt{10}ft$ $\left(iii\right)$ $2\sqrt{2}ft$
4. $\left(i\right)$ $\frac{2}{3}\sqrt{10}ft$ $\left(ii\right)$ $\frac{2}{3}\sqrt{10}ft$ $\left(iii\right)$ $2\sqrt{2}ft$

Q. A trolley-car starts from rest at the top of a hill as shown in the figure and moves down the curved track. Determine its speed as it reaches the bottom. Assume that the work done by frictional forces is negligible.

1. $\sqrt{gh}$
2. $\sqrt{2gh}$
3. $\sqrt{3gh}$
4. $\sqrt{4gh}$

Q. The position of a body moving along $x-$ axis at time $t$ is given by $x=(t^2-4t+6)m$. The distance travelled by body in time interval $t=0$ to $t=3$ s is

1. $5m$
2. $7m$
3. $4m$
4. $3m$

Q. The length of second's hand in a watch is 1 cm. The change in velocity of its tip in 15 seconds is

1. $\frac{\pi }{30\sqrt{2}}$ cm /sec
2. $\frac{\pi }{30}$ cm/sec
3. $\frac{\pi \sqrt{2}}{30}$ cm /sec
4. Zero

Q.

A 350 kg boat is 12 m long and is floating without motion on still water. A boy of mass 50 kg is at one end. The boy walks to the other end of the boat and stops. The distance in meters moved by the boy with respect to the shore is

1. 1.7

2. 1.5

3. 10.5

4. 10.3

Q. A cyclist moves from a certain point X and goes round a circle of radius 'r' and reaches Y, exactly at the other side of the point.Y, as shown in Fig.2.9. The displacement of the cyclist would be equal to :

1. $\pi r$
2. $2\pi r$
3. $2r$
4. $2\pi /r$

Q.

What are the initial position vector ${\overrightarrow{r}}_i$ and final position vector ${\overrightarrow{r}}_f$, both in unit-vector notation? What is the x component of displacement $\Delta \overrightarrow{r}$?

(i)	Position vector ${\overrightarrow{r}}_i$	(x)	$5\hat{i}-3\hat{j}-1\hat{k}$
(ii)	Postion vector ${\overrightarrow{r}}_f$	(y)	$-2\hat{i}-4\hat{j}+1\hat{k}$
(iii)	x-component of displacement $\Delta \overrightarrow{r}$	(z)	$7\hat{i}+1\hat{j}-2\hat{k}$

1. (i) - (x); (ii) - (y); (iii) - (z)

2. (i) - (z); (ii) - (y); (iii) - (x)

3. (i) - (x); (ii) - (z); (iii) - (y)

4. (i) - (z); (ii) - (x); (iii) - (y)

Q. In a two dimensional motion, instantaneous speed $v_0$ is a positive constant. Then which of the following are necessarily true?

1. The acceleration of the particle is zero
2. The acceleration of the particle is bounded
3. The acceleration of the particle is necessarily in the plane of
   motion
4. The particle must be undergoing a uniform circular motion

Q. A boy walks uniformally along the sides of a rectangular park of size 400 m $\times$ 300 m, starting from one corner to the other corner diagonally opposite. Which of the following statement is incorrect

1. He has travelled a distance of 700 m
2. His displacement is 700 m
3. His displacement is 500 m
4. His velocity is not uniform throughout the walk

Q. Find the displacement of a body as it travels to diametrically opposite point along a circle of radius 2 m.

1. 4 m
2. 3 m
3. 2 m
4. None of these

Q.

(i)	Position vector $\overrightarrow{A}$	(1)	$2\hat{i}+\hat{j}$
(ii)	Postion vector $\overrightarrow{B}$	(2)	$3\hat{i}+3\hat{j}$
(iii)	Displacement ${\overrightarrow{d}}_{AB}$	(3)	$5\hat{i}+4\hat{j}$
(iv)	Position vector of point of intersection of path 1 & path 2	(4)	$4\hat{i}+3\hat{j}$

1. (i) - (2), (ii) - (3), (iii) - (1), (iv) - (4)

2. (i) - (4), (ii) - (1), (iii) - (3), (iv) - (2)

3. (i) - (3), (ii) - (1), (iii) - (2), (iv) - (4)

4. (i) - (1), (ii) - (2), (iii) - (3), (iv) - (4)

Q. A car travels due east on a level road for $30km$. It then turns due north at an intersection and travels $40km$ before stopping. Find the resultant displacement of the car.