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A jet airplane travelling at a speed of $500\text{km/h}$ ejects its products of combustion at the speed of $1500\text{km/h}$ relative to the jet plane. What is the speed of the latter with respect to an observer on the ground?

A spherical body of area $300{\text{cm}}^2$ has emissive power of $3000{\text{J/m}}^2\text{s}$. Find the amount of thermal energy radiated by the body in $10\text{s}$.

A force $\overrightarrow{F}=2x\hat{j}$ acts on a particle. The particle starts from point $(0,0)$ and moves to the point $(3,9)$ following the path $y=x^2$. The total work done by the force $\overrightarrow{F}$ on the particle is

A particle is performing SHM with an amplitude $A$ and angular frequency $\omega$. Find the displacement of the particle from the mean position where the speed of the particle becomes half of the maximum speed.

The time taken by a particle executing simple harmonic motion of time period T to move from the mean position to half the maximum displacement is

The phase difference between input and output voltages of a CE circuit is

A board is balanced on a rough horizontal semicircular log. Equilibrium is obtained with the help of addition of a weight on the board, when the board makes an angle $\theta$ with the horizontal. Coefficient of friction between the log and the board is:

An object is displaced from position vector $\overrightarrow{r_1}=(2\hat{i}+3\hat{j})$ $\text{m}$ to $\overrightarrow{r_2}=(4\hat{i}+6\hat{j})$$\text{m}$ under a force $\overrightarrow{F}=(3x^2\hat{i}+2y\hat{j})$ $\text{N}$. Find the work done.

A stone tied to a string of length L is whirled in a vertical circle with the other end of the string at the center. At a certain instant of time, the stone is at its lowest position and has a speed u. The magnitude of the change in its velocity as it reaches a position where the string is horizontal is?

If the sides of a wall having thermal conductivity $K$ are maintained at temperatures of $T_1$ and $T_2$, the temperature as a function of $x$ can be written as $T\left(x\right)=(T_2-T_1)\left(\frac{x}{L}\right)+T_1$

Find the heat flux through the wall.

Take $T_1>T_2$

Charges $Q_1$ and $Q_2$ are in vacuum initially. $Q_2$ experiences a net force F due to the presence of $Q_1$. They are now placed in a new medium of dielectric constant ε_rand the relative distance between them is maintained. The new net force experienced by $Q_2$ due to the presence of $Q_1$ is

When two bodies stick together after collision, the collision is said to be

A block of mass $m$ is placed on the top of another block of mass $M$ as shown in the figure. The coefficient of friction between them is $\mu$. Maximum acceleration of $M$ such that $M$ and $m$ move together is (there is no friction between ground and $M$)

A fisherman hooks an old log of wood of weight 12N and volume 1000 $c{m}^3$. He pulls the log half way out of water. The tension in the string at this instant is

A ship $A$ is moving eastwards with a speed of $20\text{km/h}$ and a ship $B$ ,$50\text{km}$ south of $A$ is moving northwards with a speed of $20\text{km/h}$. Then find the shortest distance between the ships.

In some region, the gravitational field is zero. The gravitational potential in this region

Two particles $A$ and $B$ moving in $x-y$ plane are at origin at $t=0\text{sec}$.

The initial velocity vectors of $A$ and $B$ are ${\overrightarrow{u}}_A=8\hat{i}\text{m/s}$ and ${\overrightarrow{u}}_B=8\hat{j}\text{m/s}$.

The acceleration of $A$ and $B$ are constant and are ${\overrightarrow{a}}_A=-2\hat{i}{\text{m/s}}^2$ and ${\overrightarrow{a}}_B=-2\hat{j}{\text{m/s}}^2$.

List-I gives certain statements regarding particle $A$ and $B$.

List - II gives corresponding results in SI units.

Match the statements in List-I with corresponding results in List-II.

$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \text{(I)}& \text{The time (in seconds) at which velocity of A relative}& \text{(P)}& 16\sqrt{2}\\ & \text{to B is zero}& & \\ \text{(II)}& \text{The distance (in metres) between A and B when their}& \text{(Q)}& 8\sqrt{2}\\ & \text{relative velocity is zero}& & \\ \text{(III)}& \text{The magnitude of relative velocity of A and B at the}& \text{(R)}& 8\\ & \text{instant they are at same position}& & \\ & & \text{(S)}& 4\end{array}$

A gun fires a shell and recoils horizontally. If the shell travels with speed $\text{v}$ with respect to the barrel, the ratio of speed with which the gun recoils if (i) the barrel is horizontal (ii) inclined at an angle of ${30}^{\circ }$ with horizontal is

A vessel is partitioned in two equal halves by a fixed diathermic seperator. Two different ideal gases are filled in the left($L$) and right($R$) halves. The rms speed of the molecules in $L$ part is equal to the mean speed in the $R$ part. Then the ratio of the mass of a molecule in $L$ part to that of a molecule in $R$ part is

If a photocell is illuminated with a radiation of $1240{\text{A}}^{\circ }$, then stopping potential is found to be $8\text{V}$ . The work function of the emitter and the threshold wavelength are

The motion of a body is given by the equation $\frac{dv\left(t\right)}{dt}=6.0-3v\left(t\right)$. where v(t) is speed in m/s and t in sec. If body was at rest at t=0

In a given circuit, if RMS voltage across resistance and capacitor are 25 V and 30 V, then peak voltage across inductor will be

The intensity of gamma radiation from a given source is I. On passing through 36 mm of lead, it is reduced to $\frac{I}{8}$. The thickness of lead which will reduce the intensity to $\frac{l}{2}$ will be

The average depth of Indian Ocean is about $3000\text{m}$. The fractional compression, $\frac{\Delta V}{V}$of water at the bottom of the ocean (given that the bulk modulus of the water =$2.2\times {10}^9{\text{Nm}}^{-2}{\text{and g=10 ms}}^{-2}$, assuming density of water remains constant) is

If $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$, then the angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is

Find the magnitude of $\overrightarrow{B}$, if $\overrightarrow{B}$ makes an angle of ${53}^{\circ }$ West of North and $\overrightarrow{C}$ makes an angle of ${37}^{\circ }$ North of East and $\overrightarrow{A}$ has magnitude equal to $12$ and $\overrightarrow{C}$ has magnitude equal to $10$. (Given $\overrightarrow{A}+\overrightarrow{B}+\overrightarrow{C}=0$), [Hint $\sqrt{5184}=72$]

The position of a particle moving along the $x$-axis is expressed as $x=a{t}^3+b{t}^2+ct+d$. The initial acceleration of the particle is

If $\overrightarrow{A}=3\hat{i}+4\hat{j}$, $\overrightarrow{B}=\sqrt{3}\hat{i}-\hat{j}$ and angle between $\overrightarrow{A}$ and $\overrightarrow{B}$ is ${30}^{\circ }$. Find $|\overrightarrow{A}\times \overrightarrow{B}|$

Figure shows the force versus displacement graph of the particle. Find out the work done by the force to displace the particle from $x=0$ to $x=20\text{m}$.

An iron ball and a wooden ball of the same radius are released from the same height in vacuum. They take the same time to reach the ground. The reason for this is

The maximum value of mass of block $C$ so that neither $A$ nor $B$ moves is (Given that mass of $A$ is $100kg$ and that of $B$ is $140kg$. Pulleys are smooth and friction coefficient between $A$ and $B$ and between $B$ and horizontal surface is :($\mu =0.3,g=10m/s^2$).

An engineer has designed a road with a curve of radius $12\text{m}$. The material of the road offers a maximum coefficient of friction of $0.3$. What should be the maximum permissible speed limit on this road?

A particle $A$ is projected with an initial velocity of $60m/s$ at an angle ${30}^{\circ }$ to the horizontal. At the same time, a second particle $B$ is projected at an angle $\alpha$ to the horizontal with an initial speed of $50m/s$ from a point at a distance of $100m$ from $A$ as shown in the figure. If the particles collide in the air, find the angle of projection $\alpha$ of particle $B$.

70 calories of heat are required to raise the temperature of 2 moles of an ideal gas at constant pressure from ${30}^{\circ }Cto{35}^{\circ }C$. The amount of heat required to raise the temperature of same gas through the same range (${30}^{\circ }Cto{35}^{\circ }C$) at constant volume (R = 2 cal/mol/K)

A potentiometer wire of length 1 m and resistance $10\Omega$ is connected in series with a cell of e.m.f 2 volt with internal resistance $1\Omega$ and a resistance box including a resistance R. If the potential difference between the ends of the wire is 1 mV, then the value of R will be

An ideal monoatomic gas is taken round the cycle ABCA as shown in the Fig. 17.9 The work done during the cycle is

A balloon is moving vertically with a velocity of $4\text{m/s}$. When it is at a height of $h$, a ball is gently released from it. If it reaches the ground in $4\text{s}$, the height of the balloon when the body is released is

A plane is in a level flight at a constant speed and each of its two wings has an area of $50{\text{m}}^2$. If the speed of air is $180\text{km/h}$ over the lower wing surface and $252\text{km/h}$ over the upper wing surface, the plane's mass is [Lift force on the wings = weight of plane]

(Take density of air = $1{\text{kg/m}}^3$ and $g=10{\text{m/s}}^2$)

Mr. $A,B\text{and}C$ are trying to put a heavy piston into a cylinder at a mechanical workshop in a railway yard. If they apply forces $F_1,F_2\text{and}{F}_3$ respectively on the ropes, then, what is the relation between the forces if piston decends vertically with uniform speed?

An ideal gas at ${27}^{\circ }\text{C}$ is compressed adiabatically to $\frac{4}{9}$ of its original volume. If $\gamma =1.5$, then the final temperature of the gas is

A trolley of mass M oscillates with some time period about its mean position when it is connected with an ideal spring of stiffness constant K.

If an unknown mass m is gently placed at the top of trolley, its time period is found to be T.Calculate the coefficient of static friction so that the upper mass does not slip even for sizeable amplitude A.

A car is travelling along a circular curve that has a radius of $50\text{m}$ with a constant tangential acceleration. If the speed of the car changes from $36\text{km/hr}$ to $45\text{km/hr}$ in $5\text{seconds}$, the angular acceleration of the car is.(No slipping between road and car tyre)

The pressure on a square plate is measured by measuring the force on the plate and the length of the sides of the plate. If the maximum error in measurement of force and length are $4\%$ and $2\%$ respectively, then the maximum percentage error in the measurement of pressure is

Two boats $A$ and $B$ are moving along perpendicular paths in a still lake at night. At $t=0$, the boats are $300\text{m}$ apart. If boat $A$ moves with a speed of $3\text{m/s}$ and boat $B$ moves with a speed of $4\text{m/s}$ in a direction such that they can collide after sometime, then the boats will collide after

A lead ball strikes a wall and falls down but a tennis ball having the same mass and velocity striking the wall, bounces back. Check the correct statement.

Twelve wires, each having equal resistance of 2 Ω, are joined to form a cube. What is the equivalent resistance between diagonally opposite points?

Two SHM’s are represented by the equations, $y_1=0.1sin(100\pi t+\frac{\pi }{3})$ and $y_2=0.1cos100\pi t$. The phase difference between the speeds of the two particles is,