COM in Oblique Collision

Q. The escape velocity of a body on the surface of the earth is $11.2\text{km/s}$. If the earth's mass increases to twice its present value and radius of the earth becomes half, the escape velocity from the new surface becomes

1. $22.4\text{km/s}$
2. $44.8\text{km/s}$
3. $5.6\text{km/s}$
4. $11.2\text{km/s}$

Q. A modern Grand Prix racing car of mass $m$ is travelling on a flat track in a circular arc of radius $R$ with a speed $v$. If the coefficient of static friction between the tyres and the track is ${\mu }_s$, then the magnitude of negative lift $F_1$ acting downwards on the car is -

$($Assume forces on the four tyres are identical and $g=$ acceleration due to gravity$)$

1. $m(\frac{v^2}{{\mu }_{s}R}-g)$
2. $m(\frac{v^2}{{\mu }_{s}R}+g)$
3. $m(g-\frac{v^2}{{\mu }_{s}R})$
4. $-m(g+\frac{v^2}{{\mu }_{s}R})$

Q. A satellite of mass $M$ is in a circular orbit of radius $R$ about the centre of the earth. A meteorite of the same mass falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be

1. in the same circular orbit of radius $R$
2. such that it escape to infinity
3. In an elliptical orbit
4. in a circular orbit of a different radius

Q. An artificial satellite is moving in a circular orbit around the earth with speed equal to half the magnitude of escape velocity from the earth. The height of the satellite above the earth’s surface is found to be $640\times n\text{km}$. The value of $n$ will be (Give integer value).

Q. A rubber cord of a catapult has a cross sectional area $1{\text{mm}}^2$ and total unstretched length $10\text{cm}$. It is stretched to $12\text{cm}$ and then released to project a stone of mass $5\text{gm}$. Taking Young's Modulus $Y$ for rubber as $5\times {10}^8{\text{N/m}}^2$, the velocity of projection is

1. $10\text{m/s}$
2. $15\text{m/s}$
3. $20\text{m/s}$
4. $25\text{m/s}$

Q. A ball $A$ collides elastically with another identical ball $B$ with velocity $10\text{m/s}$ at an angle of ${30}^{\circ }$ with the line joining their centres $C_1$ and $C_2$. Select the correct alternative(s).

1. velocity of ball $A$ after collision is $5\text{m/s}$
2. velocity of ball $B$ after collision is $5\sqrt{3}\text{m/s}$
3. both the balls move at right angles after collision
4. kinetic energy will not be conserved here, because collision is not head on

Q. A ball of mass $m$ moving with velocity $v_0$ collides a wall as shown in figure. After impact it rebounds with a velocity $\frac{3}{4}{v}_0$. The impulse acting on ball during impact is

1. $-\frac{m}{2}{v}_0\hat{j}$
2. $-\frac{3}{4}m{v}_0\hat{i}$
3. $-\frac{5}{4}m{v}_0\hat{i}$
4. None of these

Q. A boy whirls a stone in a horizontal circle of radius $1.5$ $\text{m}$ and at a height $2.0$ $\text{m}$ above level ground. The string breaks and the stone flies off tangentially and strikes the ground after travelling a horizontal distance of $10$ $\text{m}$. What is the magnitude of centripetal acceleration of stone while in motion?

1. $163$ ${\text{m/s}}^2$
2. $64$ ${\text{m/s}}^2$
3. $15.63$ ${\text{m/s}}^2$
4. $12$ ${\text{m/s}}^2$

Q. A bob of mass m, suspended by a string of length $l_1$, is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length $l_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $\frac{l_1}{l_2}$ is

Q. A uniform rod AC of length $l$ and mass $m$ is kept on a horizontal smooth plane. It is free to rotate and move. A particle of same mass $m$ moving on the plane with velocity $v$ strikes the rod at point $\text{B}$ making an angle ${37}^{\circ }$ with the rod. The collision is elastic. After collision:

1. The angular velocity of the rod will be $\frac{72}{55}\frac{v}{l}$
   
2. The center of the rod will travel a distance $\frac{\pi l}{3}$ in the time in which it makes half rotation
3. impulse of the impact force is $\frac{24mv}{55}$
4. None of these

Q. A body $A$, of mass $m=0.1\text{kg}$ has an initial velocity of $3\hat{i}{\text{ms}}^{-1}$. It collides elastically with another body, $B$ of the same mass, which has an initial velocity of $5\hat{j}{\text{ms}}^{-1}$. After collision, $A$ moves with a velocity $\overrightarrow{v}=4(\hat{i}+\hat{j})$. The energy of $B$ after collision is written as $\frac{x}{10}\text{J}$. The value of $x$ is _____.

Q. A ball of mass $20\text{kg}$ is suspended from a massless string of length $5\text{m}$ as shown in figure. A bullet of mass $5\text{kg}$ moving with velocity $v_0\text{m/s}$ collides with the ball and sticks to it. Find the minimum value of $v_0$ so that the combined mass completes the vertical circular motion. [Take $g=10{\text{m/s}}^2$]

1. $100\text{m/s}$
2. $15\text{m/s}$
3. $79\text{m/s}$
4. $47.4\text{m/s}$

Q. Two smooth spheres made of identical material having masses $m$ and $2m$ undergo an oblique impact as shown in figure. The initial velocities of the masses are also shown. The impact force is along the line joining their centres. The coefficient of restitution is 5/9. The velocities of the masses after the impact are:

1. $\frac{10}{3}\hat{i}+8j,\frac{5}{3}\hat{i}+\frac{4}{3}j,$
2. None of these
3. $\frac{5}{3}\hat{i}-8j,\frac{-5}{3}\hat{i}+4j,$
4. $\frac{10}{3}\hat{i}-8\hat{j},\frac{-5}{3}\hat{i}+4\hat{j},$

Q. A ball of mass $m$ is suspended by a massless string of length $l=10\text{m}$ from a fixed point as shown in the figure below. A ball of mass $2m$ strikes at an angle $\theta ={45}^{\circ }$ from the horizontal and sticks to it. If the system deflects by angle $\varphi =\frac{\pi }{2}$, then the velocity of the system just after the collision and the velocity of $2m$ particle before collision are, respectively $\left(\text{in m/s}\right)$

1. $10\sqrt{2},30$
2. $10\sqrt{2},30\sqrt{2}$
3. $10,30$
4. $10,30\sqrt{2}$

Q. The point B lies on a smooth plane inclined at ${30}^{\circ }$ to the horizontal. A particle of mass $\frac{1}{7}\text{kg}$ is dropped from a point A which lies $10\text{m}$ vertically above B. The particle rebounds from the plane in the direction BC with speed $v\text{m/s}$ at an angle of ${45}^{\circ }$ to the plane. Find the impulse exerted by the plane on the particle (in $\text{Ns}$)

1. $1-\sqrt{3}$
2. $2+\sqrt{3}$
3. $2-2\sqrt{3}$
4. $1+\sqrt{3}$

Q. A solid sphere rests on a horizontal surface. A horizontal impluse $J$ is applied at height $h$ from center. The sphere starts pure rolling just after the application of the impluse. The ratio $\frac{h}{R}$ will be

1. $\frac{1}{2}$
2. $\frac{2}{5}$
3. $\frac{1}{5}$
4. $\frac{2}{3}$

Q. If the radius of earth and acceleration due to gravity on surface both are doubled, escape velocity of a satellite from surface of earth will become

1. $11.2\text{km/s}$
2. $22.4\text{km/s}$
3. $5.6\text{km/s}$
4. $44.8\text{km/s}$

Q. A plane is pulling out for a dive at a speed $720\text{km/h}$. Assuming its path to be vertical circle of radius $1000\text{m}$ and its mass to be $15000\text{kg}$, find the force exerted by the air on it at the lowest point.
(Take $g=10{\text{m/s}}^2$)

1. $500\text{kN}$
2. $750\text{kN}$
3. $147\text{kN}$
4. $600\text{kN}$

Q. A ball $A$ collides elastically with another identical ball $B$ which is at rest initially with velocity $10\text{m/s}$ at an angle of ${30}^{\circ }$ with the line joining their centres $C_1$ and $C_2$. Select the correct alternative(s).

1. velocity of ball $A$ after collision is $5\text{m/s}$
2. velocity of ball $B$ after collision is $5\sqrt{3}\text{m/s}$
3. both the balls move at right angles after collision
4. kinetic energy will not be conserved here, because collision is not head on

Q. A tunnel is dug in the earth across one of its diameters. Two masses $m$ and $2m$ are dropped from the ends of the tunnel. If the masses collide and stick to each other and perform $\text{S.H.M.}$ then amplitude of $\text{S.H.M.}$ will be: $(R=$ radius of the earth$)$

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

Q. A particle of mass $m$ moving in the $x-$ direction with speed $2v$ is hit by another particle of mass $2m$ moving in the $y-$ direction with speed $v$. If the collision is perfectly inelastic, the percentage loss in energy during the collision is close to

1. $44\%$
2. $50\%$
3. $56\%$
4. $62\%$

Q. A body of $1\text{kg}$ makes an elastic collision with a second body at rest and continues to move in the original direction, but with half of its original speed. What is the mass of the second body?

1. $\frac{1}{2}\text{kg}$
2. $2\text{kg}$
3. $\frac{1}{3}\text{kg}$
4. $3\text{kg}$

Q.

Two putty balls of equal masses moving with equal velocity in mutually perpendicular directions, stick together after collision. If the balls were initially moving with a velocity of $45\sqrt{2}m{s}^{-1}$ each, the velocity of their combined mass after collision is

1. 

2. 

3. 

4. 

Q. A ball of mass $m$ moving with velocity $v$, collide with the wall elastically as shown in the figure. After impact the change in angular momentum about $P$ is :

1. $2mvd$
2. $2mvd\cos \theta$
3. $2mvd\sin \theta$
4. $Zero$

Q. A particle start moving from rest state along a straight line under the action of a constant force and level distance $x$ first $5seconds$. The distance traveled by it in next five second will be:

1. $x$
2. $2x$
3. $3x$
4. $4x$

Q. A bob of mass $m$, suspended by a string of length $l_1$ is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass $m$ suspended by a string of length $l_2$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $l_1/l_2$ is :

Q. A particle of mass $m$ moving eastward with a speed $v$ collides with another particle of the same mass moving northwards with the same speed. If two particles coalesce on collision, the new particle of mass $2m$ will move in the north-east direction with a velocity :

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

Q. A ball $A$ collides elastically with another identical ball $B$ which is at rest initially with velocity $10\text{m/s}$ at an angle of ${30}^{\circ }$ with the line joining their centres $C_1$ and $C_2$. Select the correct alternative(s).

1. velocity of ball $A$ after collision is $5\text{m/s}$
2. velocity of ball $B$ after collision is $5\sqrt{3}\text{m/s}$
3. both the balls move at right angles after collision
4. kinetic energy will not be conserved here, because collision is not head on

Q.

Two putty balls of equal masses moving with equal velocity in mutually perpendicular directions, stick together after collision. If the balls were initially moving with a velocity of $45\sqrt{2}m{s}^{-1}$ each, the velocity of their combined mass after collision is

1. $45\sqrt{2}m{s}^{-1}$

2. $45m{s}^{-1}$

3. $90m{s}^{-1}$

4. $22.5\sqrt{2}m{s}^{-1}$

Q. A ball $A$ collides elastically with another identical ball $B$ which is at rest initially with velocity $10\text{m/s}$ at an angle of ${30}^{\circ }$ with the line joining their centres $C_1$ and $C_2$. Select the correct alternative(s).

1. velocity of ball $A$ after collision is $5\text{m/s}$
2. velocity of ball $B$ after collision is $5\sqrt{3}\text{m/s}$
3. both the balls move at right angles after collision
4. kinetic energy will not be conserved here, because collision is not head on