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(a) Show that for a projectile the angle between
the velocity and the x-axis
as a function of time is given by

(b) Show that the
projection angle
for a projectile launched from the origin is given by

Where
the symbols have their usual meaning.

If two waves represented by $y_1=4\sin \omega t$ and $y_2=3\sin (\omega t+\frac{\pi }{3})$ interfere at a point, the amplitude of the resulting wave will be close to

A spring is compressed between two toy carts of masses $m_1$ and $m_2$. When the toy carts are released, the spring exerts on each toy cart equal and opposite forces for the same time $t$. If the coefficients of friction $\mu$ between the ground and the toy carts are equal, then the displacements of the toy carts are in the ratio

With object at $\infty$, sort the focal lengths of the convex lenses as their image distance increases.

A siren of frequency $f_0$ approaches a stationary observer and then recedes from the observer. If the velocity of source $v_s$ satisfies the condition $(v_s\ll v)$ where $v$ is the velocity of sound, then the apparent change in frequency is:



[Assume, medium is stationary]

An aeroplane is flying horizontally with a velocity of 600 km/h at a height of 1960 m. When it is vertically at a point A on the ground, a bomb is released from it. The bomb strikes the ground at point B. The distance AB is

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its center. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity ${\omega }_0$. When the tortoise moves along a chord of the platform with a constant velocity (with respect to the platform), the angular velocity of the platform $\omega$(t) will vary with time t as

A block of mass $4\text{kg}$ rests on a rough horizontal surface. If a horizontal force of $12\text{N}$ is applied on the block, then the frictional force by the surface on the block is

$({\mu }_k=0.4,{\mu }_s=0.5$ and $g=10{\text{m/s}}^2)$

A block of ice at temperature $-{20}^{\circ }\text{C}$ is slowly heated and converted to steam at ${100}^{\circ }\text{C}$. Which of the following diagrams is most appropriate?

A body of mass $25\text{kg}$ slides down a smooth incline as shown. Find the work done by force of gravity till the body reaches the bottom of the inclined plane ($g=10{\text{ms}}^{-2}$)

1. 3.21

For the circuit shown $R_1=1\Omega ,R_2=2\Omega ,E_1=2\text{V}$ and $E_2=E_3=4\text{V}$. The potential difference between the points $a$ and $b$ is approximately (in $\text{Volts}$)

A charge $Q$ is placed at a perpendicular distance $\frac{a}{2}$ above the centre of the horizontal, square surface of edge length $a$ as shown in figure. Find the flux of the electric field through the square surface.

Consider the following two statements:
(A) Linear momentum of the system remians constant.
(B) Centre of mass of the system remains at rest.

A person travels along a straight road for half the distance with velocity $v_1$ and the remaining half distance with velocity $v_2$ The average velocity is given by

Find the equivalent capacitance of the system between points $\text{A}$ and $\text{B}$. All the capacitors have equal capacitance $C$.

Two particles of masses$m_1$ and $m_2$ are joined by a light rigid rod of length r. The system rotates at an angular speed

$\omega$about an axis through the centre of mass of the system and perpendicular to the rod. Show that the angular momentum of the system is L =$\mu r^2\omega$ where $\mu$ is the reduced mass of the system defined as $\mu =\frac{m_1+m_2}{m_1+m_2}$

1. 256.25

On seeing enemy ships at a distance of $125\text{m}$ from the shore, the canons start firing fireballs. The captain of the ship saw the cannon fire. He knows the velocity of the fireballs is $50\text{m/s}$. Now in how much time does he have to steer his ship in some other direction so as to save his crew. Use angle of projection as ${30}^{\circ }$. Also find the maximum height that the fireball reaches above water.

Two coaxial cylinders made of glass are immersed in a liquid of surface tension $‘S^{\prime }$. Radii of inner and outer cylinders are $R_1$ and $R_2$ respectively. Height till which the liquid will rise between the walls of the cylinder is:- (Density of liquid is $\rho$ and assume angle of contact to be $0^{\circ }$)

A piece of wood of mass $0.03\text{kg}$ is dropped from the top of a 100 m height building. At the same time, a bullet of mass $0.02\text{kg}$ is fired vertically upward with a velocity $100{\text{ms}}^{-1}$ from the ground. The bullet after collision gets embedded in the wood. Then, the maximum height to which the combined system reaches above the top of the building before falling below is $\text{Take, g}=10m{s}^{-2})$

A player kicks a football with an initial speed of $25{\text{ms}}^{-1}$ at an angle of ${45}^{\circ }$ from the ground. What are the maximum height and the time taken by the football to reach at the highest point during motion ? $(\text{Take}g=10{\text{ms}}^{-2})$

A thin uniform rod $AB$ of mass $m$ undergoes pure translation with an acceleration $a$ towards right when two antiparallel forces act on it as shown in the figure. If the distance between $F_1$ and $F_2$ is $b$, then the length of the rod is

A thin uniform straight rod of mass 2 kg and length 1 m is free to rotate about its upper end when at rest. It receives an impulsive blow of 10 Ns at its lowest point, normal to its length as shown in figure. The kinetic energy of rod just after impact is

In the connection shown in the figure the switch K is open and the capacitor is uncharged. Then we close the switch and let the capacitor charge up to the maximum and open the switch again. Then (Use the following data : $V_0=30V,R_1=10k\Omega ,R_2=5k\Omega )$

A neutron, a proton, an electron and an $\alpha$ particle enter a region of constant magnetic field with equal velocity. The $\overrightarrow{B}$ is along the perpendicularly inward to the plane of the paper. The path followed by each particle is shown in the figure.





The path followed by $\alpha$ particle is represented by:

The position of a particle along $x-$ axis at time $t$ is given by $x=1+t-t^2\text{m}$. The distance travelled by the particle in first $2\text{sec}$ is

Two particles are projected from the same point, with the same speed $u$ such that they have the same range $R$, but different maximum heights $h_1\text{and}{h}_2$. Which of the following is correct ?