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Two balls with masses $m_1$ = 3 kg and $m_2$ = 5 kg have initial velocities $v_1$ = 5 m/s = $v_2$ in the directions as shown in figure. They collide at the origin. Find the velocity of the COM 3 seconds before collision?

If a particle moves with velocity $v\left(t\right)=(2t^2-t)m/s$, then find the velocity (in $m/s$) of particle at $t=3s$.

A plane is revolving around the earth with a speed of 100 km/hr at a constant height from the surface of earth. The change in the velocity as it travels half circle is

A uniform slender bar AB of mass m and length L supported by a frictionless pivot at A is released from rest at its vertical position as shown in figure. Calculate the magnitude of the reaction at pivot when the bar just acquires the horizontal position shown dotted.

An object of uniform density is allowed to float in water kept in a beaker. The object has triangular cross-section as shown in the figure. If the water pressure measured at the three points $A,B$ and $C$ below the object are $P_A,P_B$ and $P_C$ respectively, then:

A circular wire frame is made of two uniform semicircular wires of same radius and of mass per unit lengths $\lambda$ and $2\lambda$ as shown in figure. Which of the following options is/are true?

If a car at rest is accelerated uniformly to the speed of $144\text{km/h}$ in $20\text{s}$, distance covered by the car in $20\text{s}$ is

A ball is projected upwards from the top of a tower with a velocity of $50{\text{ms}}^{-1}$ making an angle of ${60}^{\circ }$ with the vertical. If the height of the tower is $70\text{m}$, then the ball will hit the ground in

Which of the following does not affect the motion of a moving electron?

An alternating voltage E =50 sin(100 t)V is connected to a 1 μF capacitor through an ac ammeter. What will be the reading of the ammeter?

Standing wave is produced in $10\text{m}$ long stretched string. If the string vibrates in $5$ segments and wave velocity is $20\text{m/s}$, then frequency of oscillation is

A body of mass M is attached to the lower end of a metal wire, whose upper end is fixed. The elongation of this wire is l.

A conducting straight wire $PQ$ of length $l$ is fixed along a diameter of a non-conducting ring as shown in the figure. The ring is given a pure rolling motion on a horizontal surface such that its centre of mass has a velocity $v$. There exists a uniform horizontal magnetic field $B$ in horizontal direction perpendicular to the plane of ring. The magnitude of induced emf in the wire $PQ$ at the position shown in the figure will be

Two solid rubber balls A and B having masses 200 and 400 gm respectively are moving in opposite directions with velocity of A equal to 0.3 m/s. After collision the two balls come to rest, then the velocity of B is

A particle is thrown with a velocity of $um/s$. It passes $A$ and $B$ as shown in the figure at time $t_1=1s$ and $t_2=3s$. The value of $u$ is

$(g=10m/s^2)$

Find the angular displacement of the hour hand of a normal clock in $4\text{h}$

The space between the cornea and the lens of the eye is called ______ .

The position of the particle is given by $x=2t^2+6t+2$ in $\text{m}$. The instantaneous speed of the particle at $6\text{sec}$ is

The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom. If the coefficient of friction between the block and lower half of plane is given by

A box ${}^{\prime }{A}^{\prime }$ is lying on the horizaontal floor of the compartment of a train running along horizontal rails from left to right. At time ${}^{\prime }{t}^{\prime }$, it decelerates. then the reaction $R$ by the floor on the box is given best by

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II.

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ i& G{M}_e{M}_S& \text{a.}& \left(kg\right)\left(metre{)}^2\right(second{)}^{-2}\\ ii& 3RT& \text{b.}& \left(farad\right)\left(volt{)}^2\right(kg{)}^{-1}\\ iii& F^2/q^2{B}^2& \text{c.}& \left(volt\right)\left(coulomb\right)\left(metre\right)\end{array}$



Where G is universal gravitational constant; $M_e$= mass of the earth; $M_s$= mass of sun; $R$ = universal gas constant; $T$ = absolute temperature; $F$ = force; $q$ = charge; $B$ = magnetic field $\left(M{A}^{-1}{T}^{-2}\right)$. $(1\text{volt}=\frac{1\text{joule}}{1\text{coulomb}};1\text{farad}=\frac{1\text{coulomb}}{1\text{volt}})$

The coefficient of restitution 'e' for a perfectly elastic collision is

Dimensions of impulse are

A proton, a deuteron and an $\alpha$ - particle having same momentum enters a region of uniform magnetic field. Relation between radius of there trajectories is best given by

A particle is executing SHM along a straight line. Its velocities at distances $x_1$ and $x_2$ from the mean position are $v_1$ and $v_2$ respectively. Its time period of oscillation is

A particle is executing SHM. Find the phase constants of the particle, when it starts from the points $\text{A},\text{B}$ and $\text{C}$ respectively as shown below.

(Phase constant $\varphi$ is the range of $[0,2\pi ]$)

The current i in the circuit is





[IIT-JEE 1983]

The following spring block systems having same mass of the block and same spring constants were stretched or compressed to different lengths (x) and released. Match the following:

An elevator which is moving with an upward acceleration of "a $m/s^2$”. A person drops a coin to measure the acceleration 'a $m/s^2$'. Find 'a' if the coin took 0.5s to hit the ground and the coin was dropped from a height of 3 m.

A vertical column 50 cm long at ${50}^{c}irc$ C balances another column of same liquid 60 cm long at ${100}^{c}irc$°C. The coefficient of absolute expansion of the liquid is

A convex mirror of focal length f is placed at the origin with its reflecting surface towards the negative x axis. Chose the correct graph between v and u for $0\le u\le -\infty$

[The object is real

A heavy particle is projected with a velocity at an angle with the horizontal into a uniform gravitational field. The slope of the trajectory of the particle varies not according to which of the following curve?

Two cars, initially at a separation of $12\text{m}$, start simultaneously. First car $A$, starting from rest moves with an acceleration $2{\text{m/s}}^2$ , whereas the car $B$, which is ahead, moves with a constant velocity $1\text{m/s}$, away from car $A$ along the same direction. Find the time when car $A$ overtakes car $B$.

To a man walking at the rate of $3\text{km/h}$ the rain appears to fall vertically. When he increases his speed to $6\text{km/h}$, it appears to meet him at an angle of ${45}^{\circ }$ with vertical. Find the speed of rain.

Stress generated in a wire when force $F_1$ acts on it as shown in the figure, is $T$. Initial cross sectional area of the wire is $A_1$. When force $F_2$ replaces $F_1$, cross-sectional area becomes $A_2$. Find $\left(\frac{A_2}{A_1}\right)$ if $F_2=6\text{N}$.

[Consider stress generated in the wire to be the same]

The gravitational potential energy of a body of mass 'm' at the earth's surface $-mg{R}_e$. Its gravitational potential energy at a height $R_e$ from the earth's surface will be (Here $R_e$ is the radius of the earth)

Find electric field at point (2, 1) due to an electron placed at the origin

The position vector of a particle is determined by the expression

$\overrightarrow{r}=3t^2\hat{i}+4t^2\hat{j}+7\hat{k}$. The distance traversed in first 10 sec is

A boat, which has a speed of $5\text{km/h}$ in still water, crosses a river of width $1\text{km}$ along the shortest possible path in $15$ minutes. The velocity of the river water in kilometers per hour is

A uniform disc of radius r is to be suspended through a small hole made in the disc. Find the minimum possible time period of the disc for small oscillations. What should be the distance of the hole from the centre for it to have minimum time period?

Two wires are made of the same material and have the same volume. The first wire has a cross-sectional area $1{\text{mm}}^2$ and the second wire has cross-sectional area $4{\text{mm}}^2$. If the length of the first wire is increased by $\Delta L$ on applying a force of $10\text{N}$, how much force is needed to stretch the second wire by the same amount?

Two PN-junctions can be connected in series by three different methods as shown in the figure. If the potential difference in the junctions is the same, then the correct connections will be [IIT-JEE 1989]

Equation of a progressive wave is given by $y=0.2\text{cos}\pi (0.04t+0.02x-\frac{\pi }{6})$

where distances are in $\text{centimeter}$ and time in $\text{seconds}$. What will be the minimum distance between two particles having phase difference of $\frac{\pi }{2}$?

A particle ís executing simple harmonic motion with amplitude of $0.1\text{m}$. At a certain instant when its displacement is $0.02\text{m}$, its acceleration is $0.5{\text{ms}}^{-2}$. The maximum velocity of the particle is (in ${\text{ms}}^{-1}$)

A ball falls on an inclined plane of inclination $\theta ={30}^o$ from a height $h=10\text{m}$ above the point of impact and makes a perfectly elastic collision. Find the maximum height attained by the ball, perpendicuar to wedge surface. (take $g=10{\text{m/s}}^2$)

Force of gravity is least at

A brass rod of length $1\text{m}$ is fixed to a vertical wall at one end, with the other end kept free to expand. When the temperature of the rod increases by ${120}^{\circ }\text{C}$, the length increases by $3\text{cm}$. What is the strain?