Conservation of Momentum

Q. Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9M$ respectively have a centre to centre separation $8R$ as shown in the figure. A satellite of mass ${}^{\prime }{m}^{\prime }$ is projected from the surface of the planet of mass ${}^{\prime }{M}^{\prime }$ directly towards the centre of the second planet. The minimum speed ${}^{\prime }{v}^{\prime }$ required for the satellite to reach the surface of the second planet is $\sqrt{\frac{a}{7}\frac{GM}{R}}$ then the value of ${}^{\prime }{a}^{\prime }$ is _____.

[Given : The two planets are fixed in their position]

Q. The polonium isotope ${}_{84}^{210}Po$ is unstable, it emits an $\alpha$ - particle with energy $5.30\text{MeV.}$ The atomic mass of ${}_{84}^{210}Po$ is $209.9829u$ and atomic mass of ${}_2^{4}He$ is $4.0026u.$ Then,

1. Daughter nucleus is ${}_{82}^{206}Pb$
2. Disintegration energy of $\alpha$ - decay is $5.30\text{MeV}$
3. Disintegration energy of $\alpha$ - decay is $5.40\text{MeV}$
4. Mass of daughter nucleus is $205.9745u$

Q. A particle of mass $m$ is placed on top of a smooth hemispherical wedge of mass $4m$ and radius $R$ at rest. The hemispherical wedge is kept on a frictionless horizontal surface. The particle is given a gentle push. The angular velocity of the particle relative to the wedge, if wedge has velocity $v$ when particle has falls by an angle $\theta$ with respect to the wedge is

1. $\frac{5v}{R\cos \theta }$
2. $\frac{5v}{2R\cos \theta }$
3. $\frac{4v}{R\cos \theta }$
4. $\frac{5v}{R\sin \theta }$

Q. A man is standing on a cart of mass double the mass of the man. Initially cart is at rest on the smooth ground. Now man jumps with relative velocity ${}^{\prime }{v}^{\prime }$ horizontally towards right with respect to cart. Find the work done by man during the process of jumping.

1. $\frac{m{v}^2}{3}$
2. $m{v}^2$
3. $\frac{m{v}^2}{6}$
4. none of the above

Q.

Directions: Some questions (Assertion – reason type) are given below. Each question contains Statement I (Assertion) and statement II (Reason). Each question has 4 choices (a), (b), (c) and (d) out of which only one is correct. So, select the correct choice.

Statement I A single particle is having non- zero kinetic energy then its momentum can be zero.

Statement II Kinetic energy is a scalar quantity and is always positive, while momentum is a vector quantity.

1. Statement I is true, statement II is true, statement II is a correct explanation for statement I

2. Statement I is true, statement II is true, statement II is not a correct explanation for statement I

3. Statement I is true, statement II is false

4. Statement I is false, statement II is true

Q. A man of mass $80\text{kg}$ jumps from a trolley of mass $20\text{kg}$ standing on a smooth surface, with an absolute velocity of $3\text{m/s}$. The recoiling speed of the trolley is

1. $12\text{m/s}$
2. $10\text{m/s}$
3. $8\text{m/s}$
4. $11\text{m/s}$

Q. A bullet of mass $50\text{gm}$ is fired from below onto a bob of mass $450\text{gm}$ of a long simple pendulum as shown in the figure. The bullet remains inside the bob and the bob rises through a height of $1.8\text{m}$ vertically upwards. Find the speed of the bullet.

1. $60\text{m/s}$
2. $80\text{m/s}$
3. $40\text{m/s}$
4. $20\text{m/s}$

Q. A 0.5 kg ball moving with a speed of $12\text{m/s}$ strikes a rigid wall at an angle of ${30}^{\circ }$ with the wall. It is reflected with the same speed at the same angle. If the ball is in contact with the wall for 0.25 second, the average force acting on the wall is

1. 96 N
2. 48 N
3. 24 N
4. 12 N

Q. A block of mass $m$ is pushed towards a movable wedge of mass $nm$ and height $h$, with a velocity $u$. All surfaces are smooth. The minimum value of $u$ for which the block reaches the top of the wedge is -

1. $\sqrt{2gh}$
2. $2ngh$
3. $\sqrt{2gh(1+\frac{1}{n})}$
4. $\sqrt{2gh}(1-\frac{1}{n})$

Q. The polonium isotope ${}_{84}^{210}Po$ is unstable, it emits an $\alpha$ - particle with energy $5.30\text{MeV.}$ The atomic mass of ${}_{84}^{210}Po$ is $209.9829u$ and atomic mass of ${}_2^{4}He$ is $4.0026u.$ Then,

1. Daughter nucleus is ${}_{82}^{206}Pb$
2. Disintegration energy of $\alpha$ - decay is $5.30\text{MeV}$
3. Disintegration energy of $\alpha$ - decay is $5.40\text{MeV}$
4. Mass of daughter nucleus is $205.9745u$

Q. A small block of mass $m$ is placed on a body of mass $M$ at rest. If the block is set in horizontal motion with velocity $v$ as shown,

(Assume that the small block does not leave surface of $M$ and all surfaces are smooth)

1. the block $M$ will start moving due to force exerted by $m$
2. the block $M$ stays at rest as surface is smooth
3. block $M$ and $m$ always move with same speed
4. block $M$ and $m$ can never move with same speed

Q. Suppose two planets (spherical in shape) of radii $R$ and $2R$, but mass $M$ and $9M$ respectively have a centre to centre separation $8R$ as shown in the figure. A satellite of mass ${}^{\prime }{m}^{\prime }$ is projected from the surface of the planet of mass ${}^{\prime }{M}^{\prime }$ directly towards the centre of the second planet. The minimum speed ${}^{\prime }{v}^{\prime }$ required for the satellite to reach the surface of the second planet is $\sqrt{\frac{a}{7}\frac{GM}{R}}$ then the value of ${}^{\prime }{a}^{\prime }$ is _____.

[Given : The two planets are fixed in their position]

Q. A blast breaks a body initially at rest of mass $0.5\text{kg}$ into three pieces, two smaller pieces of equal mass and the third double the mass of either of small piece. After the blast the two smaller masses move at right angles to one another with equal speed. Find the statements that is/are true for this case assuming that the energy of blast is totally transferred to masses.

1. All the three pieces share the energy of blast equally
2. The speed of bigger mass is $\sqrt{2}$ times the speed of either of the smaller mass
3. The direction of motion of bigger mass makes an angle of ${135}^{\circ }$ with the direction of smaller pieces
4. The bigger piece carries double the energy of either piece.

Q. A man is standing on a cart of mass double the mass of the man. Initially cart is at rest on the smooth ground. Now man jumps with relative velocity ${}^{\prime }{v}^{\prime }$ horizontally towards right with respect to cart. Find the work done by man during the process of jumping.

1. $\frac{m{v}^2}{3}$
2. $m{v}^2$
3. $\frac{m{v}^2}{6}$
4. none of the above

Q. Initially a man and a plank of the same mass $m$ are moving on smooth horizontal surface with a velocity $v\text{m/s}$ along positive x-axis. After some time man jumps along negative x-axis with a velocity $v\text{m/s}$ with respect to ground, then the speed of the plank will be

1. $3v\text{m/s}$
2. $v\text{m/s}$
3. $2v\text{m/s}$
4. $\text{None of these}$

Q.

A missile of mass 1 kg is moving with speed 10 m/s along the +ve x direction. Its target is a car weighing 100 kg, moving with 1 m/s also in the +ve x direction. The missile hits the target but doesn't burst. Find the combined velocity of car-missile system?

1. 1 m/s

2. 10 m/s

3. $\frac{110}{101}m/s$

4. $\frac{101}{110}$ m/s

Q. Figure shows a block $A$ of mass $6m$ having a smooth semicircular groove of radius $a$ placed on a smooth horizontal surface. A block $B$ of mass $m$ is released from rest from horizontal position as shown in the figure. Find the speed of the bigger block $6m$, when the smaller block reaches the bottommost position of the groove. Take $g=10{\text{m/s}}^2$.

1. $\sqrt{\frac{ga}{14}}$
2. $\sqrt{\frac{ga}{21}}$
3. $\sqrt{\frac{ga}{10}}$
4. $\sqrt{\frac{ga}{36}}$

Q.

A wagon weighing 1000 kg is moving with a velocity 50 km/h on smooth horizontal rails. A mass of 250 kg is dropped into it. The velocity with which it moves now is

1. 12.5 km/hour

2. 20 km/hour

3. 40 km/hour

4. 50 km/hour

Q. A small block of mass $m$ is placed on a body of mass $M$ at rest. If the block is set in horizontal motion with velocity $v$ as shown,

(Assume that the small block does not leave surface of $M$ and all surfaces are smooth)

1. the block $M$ will start moving due to force exerted by $m$
2. the block $M$ stays at rest as surface is smooth
3. block $M$ and $m$ always move with same speed
4. block $M$ and $m$ can never move with same speed

Q. STATEMENT A- Linear momentum of a system of particles is zero.
STATEMENT B- Kinetic energy of system of particles is zero.

1. A does not imply B and B does not imply A.
2. A implies B but B does not imply A
3. A does not imply B but B implies A
4. A implies B and B implies A.

Q. A man (mass$=70\text{kg}$) and his son (mass$=20\text{kg}$) are standing on a frictionless surface facing in same direction. The man pushes his son, so that he starts moving with a speed of $0.90{\text{ms}}^{-1}$ with respect to man. The speed of man with respect to the surface is

1. $0.20{\text{ms}}^{-1}$
2. $0.5{\text{ms}}^{-1}$
3. $0.67{\text{ms}}^{-1}$
4. $0.85{\text{ms}}^{-1}$

Q. 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

1. $1$
2. $\frac{2}{\sqrt{3}}$
3. $\frac{\sqrt{3}}{2}$
4. $\frac{1}{2}$

Q. A riffle of mass $25\text{kg}$ fires at the rate of 6 bullets per second. If the mass of each bullet is $30\text{g}$ and the velocity of the bullet when it leaves the gun is $1080\text{km/h}$, find the magnitude of average force required to hold the gun while firing (neglect recoil speed of riffle).

1. $40\text{N}$
2. $45\text{N}$
3. $54\text{N}$
4. $60\text{N}$

Q. A man of mass $80\text{kg}$ jumps from a trolley of mass $20\text{kg}$ standing on a smooth surface, with an absolute velocity of $3\text{m/s}$. The recoiling speed of the trolley is

1. $12\text{m/s}$
2. $10\text{m/s}$
3. $8\text{m/s}$
4. $11\text{m/s}$

Q. A bullet of mass $0.01\text{kg}$ is fired from a gun weighing $2\text{kg}$. Calculate the velocity with which the gun recoils if $10$ bullets are fired at the same time with a speed of $250\text{m/s}$. Consider $+vex$ axis along the direction of firing.

1. $-12.5\text{m/s}$
2. $-1.25\text{m/s}$
3. $12.5\text{m/s}$
4. $1.25\text{m/s}$

Q. A trolley of mass $\text{M}=150\text{kg}$ is at rest over a smooth horizontal surface. Two boys each of mass $\text{m}=50\text{kg}$ are standing over the trolley. If they jump from the trolley with a relative velocity ${\text{v}}_r=20\text{m/s}$ (relative to the velocity of trolley just after jumping) one after other in same direction. The final recoil speed of trolley is

1. 6 m/s
2. 8 m/s
3. 9 m/s
4. 12 m/s

Q. A riffle of mass $25\text{kg}$ fires at the rate of 6 bullets per second. If the mass of each bullet is $30\text{g}$ and the velocity of the bullet when it leaves the gun is $1080\text{km/h}$, find the magnitude of average force required to hold the gun while firing (neglect recoil speed of riffle).

1. $40\text{N}$
2. $45\text{N}$
3. $54\text{N}$
4. $60\text{N}$

Q. Figure shows a block $A$ of mass $6m$ having a smooth semicircular groove of radius $a$ placed on a smooth horizontal surface. A block $B$ of mass $m$ is released from rest from horizontal position as shown in the figure. Find the speed of the bigger block $6m$, when the smaller block reaches the bottommost position of the groove. Take $g=10{\text{m/s}}^2$.

1. $\sqrt{\frac{ga}{14}}$
2. $\sqrt{\frac{ga}{21}}$
3. $\sqrt{\frac{ga}{10}}$
4. $\sqrt{\frac{ga}{36}}$

Q. A bomb of mass $3m\text{kg}$ explodes into two pieces of mass $m\text{kg}$ and $2m\text{kg}$ . If the velocity of $m\text{kg}$ mass is $16\text{m/s}$, the total energy released in the explosion is

1. $192m\text{J}$
2. $96m\text{J}$
3. $384m\text{J}$
4. $768m\text{J}$

Q. A bomb of mass $2\text{kg}$ at rest explodes into two chunks of masses $0.5\text{kg}$ and $1.5\text{kg}$. Velocity of $0.5\text{kg}$ chunk is ($2\hat{i}+4\hat{j}+8\hat{k}$) $\text{m/s}$. Velocity of $1.5\text{kg}$ chunk is -

1. $(\frac{2}{3}\hat{i}+\frac{4}{3}\hat{j}+\frac{8}{3}\hat{k})$ $\text{m/s}$
2. $-(\frac{2}{3}\hat{i}+\frac{4}{3}\hat{j}+\frac{8}{3}\hat{k})$ $\text{m/s}$
3. $(\frac{2}{3}\hat{i}-\frac{4}{3}\hat{j}+\frac{8}{3}\hat{k})$ $\text{m/s}$
4. $(\frac{2}{3}\hat{i}-\frac{4}{3}\hat{j}-\frac{8}{3}\hat{k})$ $\text{m/s}$