Summary for Maximum Range and the Dominating Component

Q. Point masses m1 and m2 are placed at the opposite ends of a rigid rod of length L, and negligeable mass. The rod is set rotating about an axis perpendicular to it. The position of point P on this rod through which axis should pass so that the work required to set the rod rotating with angular velocity w0(omega) is minimum, is given by

Q. A fish is rising up vertically inside a pond with velocity $4\text{cm/s}$, and notices a bird, which is diving downward and its velocity appears to be $16\text{cm/s}$ (to the fish). The real velocity of the diving bird is $[{\mu }_{\text{water}}=4/3]$

1. $9\text{cm/s}$
2. $12\text{cm/s}$
3. $6\text{cm/s}$
4. $15\text{cm/s}$

Q.

Two particles are projected from the ground simultaneously with speeds $20m/s$ and $\frac{20}{\sqrt{3}}m/s$at angles ${30}^{\circ }$ and ${60}^{\circ }$ with the horizontal in the same direction. The maximum distance between them till both of them strike the ground is approximately ($g$ = $10m/s^2$)

1. 30.2 m

2. 10.4 m

3. 23.1 m

4. 16.4 m

Q. A ball is thrown from ground level so as to just clear a wall $4\text{m}$ high at a distance of $4\text{m}$ and falls at a distance of $14\text{m}$ from the wall. Find the magnitude of the initial velocity (in $\text{m/s}$) of the ball.

[Upto two decimal places]

Q. If a charged particle enters perpendicularly in the uniform magnetic field then 1. Energy Remain cons†an t but momentum changes 2. Energy and momentum both remain cons†an t 3. Momentum remain cons†an t but energy changes 4. Neither energy nor momentum remains cons†an

Q.

In the motion of a projectile freely under gravity, its

1. Momentum is conserved

2. Total energy is conserved

3. Energy and momentum both are conserved

4. None is conserved

Q. A projectile is to be launched at an angle ${45}^{\circ }$ so that it falls beyond the pond of length $13.6$ $\text{m}$ as shown in the figure and distance between both the points $O$ to $P$ and $M$ to $N$ is $6$ $\text{m}$. What is the range of values of initial velocity (in $m/s$) so that the projectile falls between points $M$ and $N$.

1. $15<u<16$
2. $14<u<16$
3. $13<u<16$
4. $13<u<14$

Q. A fish is rising up vertically inside a pond with velocity $4\text{cm/s}$ and notices a bird, which is diving downward and its velocity appears to be $16\text{cm/s}$ (to the fish). The real velocity of the diving bird is $[{\mu }_{\text{water}}=4/3]$

1. $9\text{cm/s}$
2. $12\text{cm/s}$
3. $6\text{cm/s}$
4. $15\text{cm/s}$

Q. A point sized sphere of mass m is suspended from a point using a string of length l. It is then pulledto a side till the string is horizontal and released.as the mass passes through the portion where the string is vertical, magnitude of its angular momentum is

Q. A hunter aims at a monkey sitting on a tree at a considerable distance. At the instant he fires at it, the monkey drops. Will the bullet hit the monkey?

1. Yes
2. Sometimes
3. Never
4. No

Q. A projectile is to be launched at an angle ${45}^{\circ }$ so that it falls beyond the pond of length $13.6$ $\text{m}$ as shown in the figure and distance between both the points $O$ to $P$ and $M$ to $N$ is $6$ $\text{m}$. What is the range of values of initial velocity (in $m/s$) so that the projectile falls between points $M$ and $N$.

1. $15<u<16$
2. $14<u<16$
3. $13<u<16$
4. $13<u<14$

Q. Two particles are in SHM with same angular frequency and amplitudes $A$ and $2A$ respectively along same straight line with same mean position. They cross each other at position $\frac{A}{2}$ distance from mean position in opposite direction.The phase difference between them is :

1. $\frac{5\pi }{6}-{\sin }^{-1}\left(\frac{1}{4}\right)$
2. $\frac{\pi }{6}-{\sin }^{-1}\left(\frac{1}{4}\right)$
3. $\frac{5\pi }{6}-{\cos }^{-1}\left(\frac{1}{4}\right)$
4. $\frac{\pi }{6}-{\cos }^{-1}\left(\frac{1}{4}\right)$

Q. Two particles projected from the same point with same speed u at angles of projection $\alpha$ and $\beta$ strike the horizontal ground at the same point. If $h_1$ and $h_2$ are the maximum heights attained by projectiles, $R$ be the range for both and $t_1$ and $t_2$ be their time of flights respectively, then

1. $\alpha +\beta =\frac{\pi }{2}$
2. $\frac{t_1}{t_2}=tan\alpha$
3. $R=4\sqrt{h_1{h}_2}$
4. $tan\alpha =\sqrt{\frac{h_1}{h_2}}$

Q.

In the motion of a projectile freely under gravity, its

1. Total energy is conserved

2. Momentum is conserved

3. Energy and momentum both are conserved

4. None is conserved

Q. A player kicks a football obliquely at a speed of $20m/s$ so that it's range is maximum. Another player at a distance of $24m$ away in the direction of kick, starts running at that instant to catch the ball. The constant speed with which he has to run is $(g=10m{s}^{-2})$

1. $4m{s}^{-1}$
2. $4\sqrt{2}m{s}^{-1}$
3. $8\sqrt{2}m{s}^{-1}$
4. $8m{s}^{-1}$

Q. Two particles are simultaneously projected in the horizontal direction from a point $P$ at a certain height. The initial velocities of the particles are oppositely directed to each other and have magnitude v each. The separation between the particles at a time when their position vectors (drawn from the point P) are mutually perpendicular, is

1. $\frac{v^2}{2g}$
2. $\frac{4v^2}{g}$
3. $\frac{v^2}{g}$
4. $\frac{2v^2}{g}$

Q. Two stones are projected so as to reach the same distance from the point of projection on a horizontal surface. The maximum height reached by one exceeds the other by an amount equal to half the sum of the heights attained by them. Then the angles of projection for the stones are (in degrees)

1. $30,60$
2. $0,90$
3. $70,20$
4. $80,10$

Q. Two particles $A$ and $B$ are projected from ground towards each other with speeds $10m/s$ and $5\sqrt{2}m/s$ at angles ${30}^{\circ }$ and ${45}^{\circ }$ with horizontal from two points separated by a distance of $15m$. Will they collide or not?

Q. A gun is held at an angle of ${45}^{\circ }$ above the horizontal.
(i) If the range of the projectile is $25m$, what is its muzzle velocity?
(ii) At what angle should the gun be held so that the range is reduced to $18m$?

Q. A particle is subjected to two SHMs simultaneously $x_1=a_1\sin \omega t$ and $x_2=a_{2}sin(\omega t+\varphi )$ Where $a_1=3.0cm$, $a_2=4.0cm$. Find resultant amplitude if the phase difference $\varphi$ has values(a) 0 (b) 60 (c) 90

Q. A small block of mass $m$, having charge $q$ is placed on frictionless inclined plane making an angle $\theta$ with the horizontal. There exists a uniform magnetic field $B$ parallel to the inclined plane but perpendicular to the length of spring. If $m$ is slightly pulled on the inclined in downward direction and released, the time period of oscillation will be (assume that the block does not leave contact with the plane):

1. $2\pi \sqrt{\frac{m}{K}}$
2. $2\pi \sqrt{\frac{2m}{K}}$
3. $2\pi \sqrt{\frac{qB}{K}}$
4. $2\pi \sqrt{\frac{qB}{2K}}$

Q. Consider the above system, in which a mass $m$ is balanced on a FRICTIONLESS incline with the help of a vertical cord exerting tension $T$ directly upwards on the mass.
What is the normal force $F_N$ exerted by the incline on $m$?
The incline makes an angle of ${30.0}^o$ with the horizontal.

1. $F_N=0$
2. $F_N=mg-T$
3. $F_N=\frac{mg-T}{\sin \theta }$
4. $F_N=mg\tan \theta$
5. $F_N=T\tan \theta$

Q. A block placed on a rough inclined plane of inclination $(\theta ={30}^0)$ can just be pushed upwards by applying a force "F" as shown. If the angle of inclination of the inclined plane is increased to $(\theta ={60}^0)$, the same block can just be prevented from sliding down by application of a force of same magnitude. The coefficient of friction between the block and the inclined plane is

1. $\frac{\sqrt{3}+1}{\sqrt{3}-1}$
2. $\frac{2\sqrt{3}-1}{\sqrt{3}+1}$
3. None of these
4. $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

Q. Find out time period of small oscillations from mean position

1. $2\pi \sqrt{\frac{l}{g}}$
2. $2\pi \sqrt{\frac{l}{{(g^2+{\left(\frac{qE}{m}\right)}^2)}^{1/2}}}$
3. $\frac{1}{2\pi }\sqrt{\frac{l}{g}}$
4. $\frac{1}{2\pi }\sqrt{\frac{l}{{(g^2+{\left(\frac{qE}{m}\right)}^2)}^{1/2}}}$

Q. A cylindrical vessel of cross-sectional area, $s$, is left out in the rain in which water is falling vertically downward with the velocity, $v$, in the still air. When the wind starts blowing in North-East direction with velocity, $v$, the rate of collection of water in the vessel is

1. $v.s$
2. $\sqrt{2}v.s$
3. $2v.s$
4. $2\sqrt{2}v.s$

Q. A body is projected vertically upwards with a velocity of $10{ms}^{-1}$ and another body is projected simultaneously from the same point with a velocity of $20{ms}^{-1}$ at an angle of $\frac{\pi }{6}$ with the horizontal. The distance between the two bodies after one second from the time of projection is (Acceleration due to gravity is $10{ms}^{-2}$)

1. $10\sqrt{3}m$
2. $10m$
3. $20m$
4. $20\sqrt{3}m$

Q. A block is kept on an inclined plane of angle ${30}^o$. The coefficient of kinetic friction between block and the inclined plane is $1\sqrt{3}$.What is the acceleration of the block ?

1. Zero
2. $2m/s^2$
3. $1.5m/s^2$
4. $5m/s^2$

Q. The upper half of an incline plane with inclination $\varphi$ is perfectly smooth while the lower half is rough A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by :

1. $2\sin \varphi$
2. $2\tan \varphi$
3. $2\cos \varphi$
4. $\tan \varphi$

Q. Match the statements in Column I with those in Column II. One or more matching is possible.

Q. Two particles are projected simultaneously in a vertical plane from the same point. These particles have different velocities at different angles with the horizontal. The path seen by each other is

1. parabola
2. elliptical
3. straight line
4. hyperbola