Instantaneous Velocity

Q.

The position vector of a particle $\overrightarrow{R}$ as a function of time is given by $\overrightarrow{R}=4\sin \left(2\pi t\right)\hat{i}+4\cos \left(2\pi t\right)\hat{j}$, 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 particle?

1. Path of the particle is a circle of radius $4$ $m$
2. Acceleration vector is along $-\overrightarrow{R}$
3. Magnitude of acceleration vector is $\frac{v^2}{R}$, where $v$ is the velocity of particle
4. Magnitude of the velocity of particle is $8$ $m{s}^{-1}$

Q.

A particle moves along a straight line $OX$. At a time $ts$, the distance $xm$ of the particle from $O$ is given by $x=40+12t–t^3$. How long would the particle travel before coming to rest?

1. $24m$

2. $40m$

3. $56m$

4. $16m$

Q. A particle is moving with speed $v=b\sqrt{x}$ along the positive $x$ - axis. Calculate the speed of the particle at time $t=\tau$. (Assume that the particle is at the origin at $t=0$).

1. $\frac{b^2\tau }{2}$
2. $b^2\tau$
3. $\frac{b^2\tau }{\sqrt{2}}$
4. $\frac{b^2\tau }{4}$

Q.

A particle is moving along a circular path of radius 5m with a uniform speed 5m per second. What will be the average acceleration when the particle completes half revolution?

Q. The kinetic energy of a projectile at the highest point is half of the initial kinetic energy. What is the angle of projection with the horizontal?

1. ${45}^{\circ }$
2. ${30}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q.

Explain the terms the average velocity and instantaneous velocity. When are they equal?

Q. A particle of mass $2\text{gm}$ and charge $1\mu \text{C}$ is held at rest on a frictionless horizontal surface at a distance of $1\text{m}$ from the fixed charge of $1\text{mC}$. If the particle is released, it will be repelled. The speed of the particle when it is at a distance of $10\text{m}$ from the fixed charge is

1. $45{\text{ms}}^{-1}$
2. $60{\text{ms}}^{-1}$
3. $90{\text{ms}}^{-1}$
4. $100{\text{ms}}^{-1}$

Q. The intensity ratio of two waves is 9 : 1. These wave produce interference. The ratio of maximum intensity to minimum intensity will be

1. 1 : 9
2. 9 : 1
3. 1 : 4
4. 4 : 1

Q. The two ends of train moving with constant acceleration pass a certain point with velocities u and 3u the velocity with which middle point of the train passes the same point

Q. A point charge $+Q$ having mass $m$ is fixed on horizontal smooth surface. Another point charge having magnitude $+2Q$ and mass $2m$ is projected horizontal towards the charge $+Q$ from far distance with velocity $v_o$.

Acceleration of the particle $+2Q$ when it is closest to fixed particle $+Q$ :

1. Zero
2. $\frac{m{v}_o^4}{2k{Q}^2}$
3. $\frac{m{v}_o^4}{4k{Q}^2}$
4. $\frac{m{v}_o^4}{k{Q}^2}$

Q.

A person traveling at 43.2 km/h applies the brake giving a deceleration of 6.0 m/$s^2$ to his scooter. How far will it travel before stopping?

Q.

The numerical ratio of average velocity to average speed is.

1. always less than one

2. always equal to one

3. always more than one

4. equal to or less than one

Q. A beam of unpolarized light of intensity $I_0$ is passed through a polaroid $A$ and then through another polaroid $B$. Polaroid $B$ is oriented so that its principal plane makes an angle of ${45}^{\circ }$ relative to that of $A$. The intensity of the emergent light is,

1. $I_0$
2. $I_0/2$
3. $I_0/4$
4. $I_0/8$

Q. A point charge $q$ having mass $m$ is projected from a long distance with speed $v$ towards another stationary particle of same mass and charge. The distance of closest approach of the particles is :

1. $\frac{q^2}{2\pi {\epsilon }_{0}m{v}^2}$

2. $\frac{2q^2}{\pi {\epsilon }_{0}m{v}^2}$

3. $\frac{q^2}{\pi {\epsilon }_{0}m{v}^2}$
4. Zero

Q.

An aeroplane flying 490 m above ground level at 100 m/s, releases a block. What horizontal distance would it travel before it hits the ground?

1. 0.1 km

2. 0.5 km

3. 1 km

4. 2 km

Q. Rest mass energy of an electron is 0.51 MeV. If this electron is moving with a velocity 0.8 c (where c is velocity of light in vacuum), then kinetic energy of the electron should be.

1. 0.28 MeV
2. 0.34 MeV
3. 0.39 MeV
4. 0.46 MeV
   

Q. Column - I gives a list of possible set of parameter measures in some experiments. The variations of the parameters in the form of graphs are shown in Column - II. Match the set of parameters given in Column - I with the graphs given in Column - II.

1. $A\to p,s;B\to q,s;C\to s;D\to q$
2. $A\to p,r;B\to q,r;C\to s;D\to q$
3. $A\to p,r;B\to q,s;C\to r;D\to q$
4. $A\to p,s;B\to q,s;C\to s;D\to p$

Q. A beam of natural light falls on a system of $5$ Polaroids, which are arranged in succession such that the pass axis of each Polaroid is turned through ${60}^{\circ }$ with respect to the preceding one. The fraction of the incident light intensity that passes through the system

1. $\frac{1}{64}$
2. $\frac{1}{32}$
3. $\frac{1}{512}$
4. $\frac{1}{128}$

Q.

A particle moves on xy plane. Its position vector at any time t is $\overrightarrow{r}=\left\{\right(2t)\hat{i}+(2t^2\left)\hat{j}\right\}m$. The rate of change of $\theta$ at time t=2 second. (Where $\theta$ is the angle which its velocity vector makes with positive x-axis) is

1. 

2. 

3. 

4. 

Q.

When the distance travelled by an object is directly proportional to the time, it is said to travel with

1. zero velocity

2. Constant speed

3. constant acceleration

4. uniform velocity

Q. A car is moving along a straight line OP . it moves O to P I n 18sec and returns from P to Q in 6sec . where OP =360m and OQ=240m. What are the average velocity and average speed of the car in going: (1) From O to P (2) O to P and back to Q.

Q.

A particle moves so that $S=6+48t-t^3$. The direction of motion reverses after moving a distance of

1. $63$

2. $104$

3. $134$

4. $288$

Q. A clock has its second hand 2.0 cm long.Find the average speed and modulus of average velocity of the second hand in 15s.

Q. What is the velocity of projectile when it mooves perpendicular to initial velocity?

Q.

If youre driving one and a half miles per minute, slow down by $15$ miles per hour, and then reduce your speed by one third, how fast are you going now?

Q. The rotation of the earth having radius $R$ about its axis speeds up to a value such that a man at latitude of ${60}^{\circ }$ feels weightlessness. The duration of the day in such a case is

1. $\pi \sqrt{\frac{R}{g}}$
2. $\frac{\pi }{2}\sqrt{\frac{R}{g}}$
3. $\frac{\pi }{3}\sqrt{\frac{R}{g}}$
4. $2\sqrt{\frac{g}{R}}$

Q.

A body starts from a point O with velocity v and acceleration a. The direction of acceleration is reversed when the velocity becomes 5v. The velocity of the body at point O will be

1. -v

2. -5v

3. -7v

4. -9v

Q. A particle moves according to the equation $x=2t^2-5t+6$ where, $x$ is in $\text{m}$. Find
$\left(i\right)$ average velocity in the first $3\text{s}$, and
$\left(ii\right)$ instantaneous velocity at $t=3\text{s}$.

1. $2\text{m/s},5\text{m/s}$
2. $1\text{m/s},7\text{m/s}$
3. $4\text{m/s},3\text{m/s}$
4. $3\text{m/s},7\text{m/s}$

Q.

Figure shows the graph of the x-coordinate of a particle going along the X-axis as a function of time. Find

(a) The average velocity during 0 to 10 s,

(b) Instantaneous velocity at 2, 5, 8 and 12s

Q. A satellite is revolving around the earth in a circular orbit of radius 'a' with velocity $v_0$. A particle is projected from the statellite in forward direction wiht relative velocity $V=\left[\right(\frac{\sqrt{5}}{4}\left)\right]v_0.$ During the subsequent motion of the particle, match the following:
$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{I. Total energy of particle}& \text{P.}\frac{(-3G{M}_{e}m)}{8a}\\ \text{II. Minimum distance of}& \text{Q.}\frac{(-5G{M}_{e}m)}{8a}\\ \text{particle from the earth}& \\ \text{III. Maximum distance of}& \text{R.}\frac{5a}{3}\\ \text{particle from the earth}& \\ & \text{S. 2a}\\ & \text{T.}\frac{2a}{3}\\ & \text{U. a}\end{array}$

1. $I\to P;II\to U;III\to S$
2. $I\to Q;II\to S;III\to R$
3. $I\to P;II\to U;III\to R$
4. $I\to Q;II\to S;III\to U$