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Question: A particle of mass $100 \mathrm{~g}$ moving at an initial speed u collides with another particle of same mass kept initially at rest. If the total kinetic energy becomes $0.2 \mathrm{~J}$ after the collision, what could be the minimum and the maximum value of $u$. Solution: Use C.O.L.M $0.1(u)=0.1\left(v_{1}+v_{2}\right)$ $\frac{1}{2} m_{1} v_{1}^{2}+\frac{1}{2} m_{2} v_{2}^{2}=0.2$ $\left(v_{1}^{2}+v_{2}^{2}\right)=\frac{0.2 \times 2}{0.1}=4$ $u=v_{1}+v_{2}$ $=\sqrt{4-v_{2}^{2}}+v_{2}...
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Question: A block of mass $2.0 \mathrm{~kg}$ moving at $2.0 \mathrm{~m} / \mathrm{s}$ collides head on with another block of equal mass kept at rest. (a) Find the maximum possible loss in kinetic energy due to the collision. (b) If the actual loss in kinetic energy is half of this maximum, find the co-efficient of restitution. Solution: (a) Use C.O.L.M $2 .(2)+0=2\left(v_{1}+v_{2}\right)$ $v_{1}+v_{2}=2$ $(K . E)_{\text {loss }}=\frac{1}{2} m\left(v_{1}^{2}+v_{2}^{2}\right)-\frac{1}{2} m(4)$ For...
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Question: A ball of $m$ moving at a speed $v$ makes a head-on collision with an identical ball at rest. The kinetic energy of the balls after the collision is three fourths of the original. Find the coefficient of restitution. Solution: Use C.O.L.M v=initial velocity $v_{1}, v_{2} \rightarrow_{\text {final velocities }}$ $m v=m\left(v_{1}+v_{2}\right) \Longrightarrow v=v_{1}+v_{2}-(1)$ Given $\frac{1}{2} m\left(v_{1}^{2}+v_{2}^{2}\right)=\frac{3}{4} \cdot \frac{1}{2} m v^{2}$ $v_{1}^{2}+v_{2}^{2...
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Question: A block of mass $200 \mathrm{~g}$ is suspended through a vertical spring. The spring is stretched by $1.0 \mathrm{~cm}$ when the block is in equilibrium. A particle of mass $120 \mathrm{~g}$ is dropped on the block from a height of $45 \mathrm{~cm}$. The particle sticks to the block after the impact. Find the maximum extension of the spring. Take $g=10 \mathrm{~m} / \mathrm{s}^{2}$. Solution: At equilibrium $k x_{0}=200 \mathrm{~g}$ (before $120 \mathrm{~g}$ falls) $k=\frac{0.200 \time...
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Question: A ball falls on an inclined plane of inclination ${ }^{\theta}$ from a height $h$ above the point of impact and makes a perfectly elastic collision. Where will it hit the plane again? Solution: After falling down, ball will have a projectile motion. $\alpha=\frac{\pi}{2}-2 \theta$ $u=\sqrt{2 g h}, O P=l$ $x=l \cos \theta \quad y=-l \sin \theta$ Equation of trajectory $\Rightarrow y=x \tan \alpha-\frac{g x^{2} \sec ^{2} \alpha}{2 u^{2}}$ $\Rightarrow-l \sin \theta=l \cos \theta \cdot \t...
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Question: A projectile is fired with a speed $u$ at an angle ${ }^{\theta}$ above a horizontal field. The coefficient of restitution of collision between the projectile and the field is e. How far from the starting point, does the projectile makes its second collision with the field? Solution: $\mathrm{e}=$ co-efficient of restitution $$ u_{x}=u \cos \theta $$ After collision with ground, $u_{y}{ }^{x}=e u \sin \theta$ $u_{x}{ }^{x}=u_{x}$ $u=\sqrt{\left(u x^{2}\right)^{2}+\left(u y^{\circ}\righ...
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Question: A bullet of mass $20 \mathrm{~g}$ travelling horizontally with a speed of $500 \mathrm{~m} / \mathrm{s}$ passes through a wooden block of mass $10.0 \mathrm{~kg}$ initially at rest on a level surface. The bullet emerges with a speed of $100 \mathrm{~m} / \mathrm{s}$ and the block slides $20 \mathrm{~cm}$ on the surface before coming to rest. Find the friction coefficient between the block and the surface. Solution: $f=m a$ $\mu N=m a$ $\mu m g=m a$ $a=\mu .10$ Use C.O.L.M $0.02(500)=10...
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Question: A block of mass $2.0 \mathrm{~kg}$ is moving on a frictionless horizontal surface with a velocity of $1.0 \mathrm{~m} / \mathrm{s}$ towards another block of mass kept at rest. The spring constant of the spring fixed at one end is $100 \mathrm{~N} / \mathrm{m}$. Find the maximum compression of the spring. Solution: We know that for the spring, $a=-\omega^{2} x$ $a=\frac{-k}{m} \times x$ $a=\frac{-100}{2} \times x=-50 x$ Use law of kinematics, $v^{2}=u^{2}+2 a x \quad\{u=1 m / s\}$ $0=1-...
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Question: In a gamma decay process, the initial energy of a nucleus of mass $M$ decreases, a gamma photon of energy $\mathrm{E}$ and linear momentum $\mathrm{E} / \mathrm{c}$ is emitted and the nucleus recoils. Find the decrease in internal energy. Solution: K.E. of nucleus $=\frac{p N^{2}}{2 M_{N}}=\frac{E^{2}}{2 M_{N} c^{2}}$ Decrease in internal energy $E_{\text {int }(N)}-E_{\text {int }\left(N^{\prime}\right)}=E_{r}+K \cdot E$ $=E+\frac{E^{2}}{2 M_{N} c^{2}}$...
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Question: A bullet of mass $m$ moving at a speed $v$ hits a ball of mass $M$ kept at rest. A small part having mass $m$ breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed of $v_{1}$ in the direction of bullet. Find the velocity of the bullet after the collision. Solution: Use C.O.L.M, $m v=\left(m^{\prime}+m\right) V_{2}+\left(M+m^{\prime}\right) V_{1}$ $V_{2}=\frac{m v-\left(M-m^{\prime}\right) V_{1}}{\left(m^{\prime}+m\right)}$...
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Question: Consider a head-on collision between two particles of masses $m_{1}$ and $m_{2}$. The initial speeds of the particles are ${ }^{u_{1}}$ and ${ }^{u_{2}}$ in the same direction. The collision starts at $\mathrm{t}=0$ and the particles interact for a time interval $\Delta t$. During the collision, the speed of the first particle varies as $v(t)=u_{1}+\frac{t}{\Delta t}\left(v_{1}-u_{1}\right)$ Find the speed of the second particle as a function of time during the collision. Solution: Use...
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Question: A $60 \mathrm{~kg}$ man skating with a speed of $10 \mathrm{~m} / \mathrm{s}$ collides with a $40 \mathrm{~kg}$ skater at rest and they cling to each other. Find the loss of kinetic energy during the collision. Solution: $m_{1}=60, v_{1}=10, m_{2}=40, v_{2}=0, V=v_{\text {final }}$ Use C.O.L.M $60 \times 10+0=(60+40) \mathrm{V}$ $V=\frac{600}{100}=6$ $\Delta K \cdot E_{\text {loss }}=\left|\frac{1}{2} \times 100 \times 6^{2}-\frac{1}{2} \times 60 \times 100\right|$ $\Delta K . E_{\text...
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Question: A ball of mass $0.50 \mathrm{~kg}$ moving at a speed of $5.0 \mathrm{~m} / \mathrm{s}$ collides with another ball of mass $1.0 \mathrm{~kg}$. After the collision the balls stick together and remain motionless. What was the velocity of the $1.0 \mathrm{~kg}$ block before the collision? Solution: Use C.O.L.M $m_{1}=0.5, v_{1}=\frac{5 m}{s}, m_{2}=1, v_{2}=?$ $0.5 \times 5+1 \times v_{2}=0$ $v_{2}=-2.5 \mathrm{~m} / \mathrm{s}$...
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Question: In a typical Indian Bugghi (a luxury cart drawn by horses), a wooden plate is fixed on the rear on which one person can sit. A bugghiof mass $200 \mathrm{~kg}$ is moving at a speed of $10 \mathrm{~km} / \mathrm{h}$. As it overtakes a school boy walking at a speed of $4 \mathrm{~km} / \mathrm{h}$, the boy sits on the wooden plate. If the mass of the boy is $25 \mathrm{~kg}$, what will be the new velocity of the bugghi? Solution: C.O.L.M (conservation of linear momentum) $m_{s} v_{s}+m_{...
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Question: Figure shows a small block of mass $m$ which is started with a speed of $v$ on the horizontal part of the bigger block of mass $M$ placed on a horizontal floor. The curved part of the surface shown is semicircular. All the surfaces are frictionless. Find the speed of the bigger block when the smaller block reaches the point $\mathrm{A}$ of the surface. Solution: Use C.O.L.M $m v=(M+m) v^{\prime \prime} \Rightarrow v^{\prime}=\frac{m v}{M+m}$...
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Question: Two persons each of mass $m$ are standing at the two extremes of a railroad car of mass $M$ resting on a smooth track. The person on left jumps to the left with the horizontal speed u with respect to the state of car before the jump. Thereafter, the other person jumps to the right, again with the same Two persons each of mass $m$ are standing at the two extremes of a railroad car of mass $M$ resting on a smooth track. The person on left jumps to the left with the horizontal speed u wit...
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Question: A gun is mounted on a railroad car. The mass of the car, the gun, the shells and the operator is $50 \mathrm{~m}$ where $\mathrm{m}$ is the mass of one shell. If the velocity of the shell with respect to the gun (in its state before firing) is $200 \mathrm{~m} / \mathrm{s}$, what is the recoil speed of the car after the second shot? Neglect the friction. Solution: After 1 bullet, C.O.L.M $\Rightarrow 0=49 m \times V_{1}+m(200)$ $\Rightarrow V_{1}=\frac{-200}{49}$ After $2^{\text {nd }}...
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Question: A railroad car of mass $M$ is at rest on frictionlessrails when a man of mass $m$ starts moving on the car towards the engine. If the car recoils with a speed $v$ backward on the rails, with what velocity is the man approaching the engine? Solution: C.OL.M $V_{1}=$ velocity with which man in car approaches. $\Rightarrow(M+m) V=m V_{1}$ $\Rightarrow V_{1}=\frac{(M+m) V}{m}=\left(1+\frac{M}{m}\right) V$...
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Question: A ball of mass $m$ is dropped onto a floor from a certain height. The collision is perfectly elastic and the ball rebounds to the same height and again falls. Find the average force exerted by the ball on the floor during a long time interval. Solution: $v^{2}=2 g h \Rightarrow v=\sqrt{2 g h}$ $v^{\prime}=-\sqrt{2 g h}$ $F=\frac{m \Delta v}{\Delta t}=\frac{m 2 \sqrt{2 g h}}{\Delta t}$ For falling down, Time taken $t_{1} \Rightarrow 0=+v+a t_{1}$ $\Rightarrow v=g t_{1} \quad\{a=-g\}$...
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Question: During a heavy rain, hailstones of average size $1.0 \mathrm{~cm}$ in diameter fall with an average speed of $20 \mathrm{~m} / \mathrm{s}$. Suppose 2000 hailstorms strike every square meter of a $10 \mathrm{~m}^{\times} 10 \mathrm{~m}$ roof perpendicularly in one second and the average force exerted by the falling hailstones on the roof. Density of a hailstone is $900^{\mathrm{kg} / \mathrm{m}^{3}}$ Solution: Volume of 1 hailstorm $=\frac{4}{3} \pi\left(\frac{10^{-2}}{2}\right)^{3}=\fr...
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Question: Two fat astronauts each of mass $120 \mathrm{~kg}$ are travelling in a closed spaceship moving at a speed of $15 \mathrm{~km} / \mathrm{s}$ in the outer space far removed from all other material objects. The total mass of the spaceship and its contents including the astronauts is $660 \mathrm{~kg}$. If the astronauts do slimming exercise and thereby reduce their masses to $90 \mathrm{~kg}$ each, with what velocity will the spaceship move? Solution: In a closed spaceship, there is no ex...
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Question: A block at rest explodes into three equal parts. Two parts starts moving along $X$ and $Y$ axes respectively with equal speeds of $10 \mathrm{~m} / \mathrm{s}$. Find the initial velocity of the third part. Solution: resultant velocity of 2 particles moving along $x$ and $y$-axis $V_{R}=\sqrt{(10)^{2}+(10)^{2}+2(10)(10) \cos 90^{\circ}}$ $=10 \sqrt{2} \mathrm{~m} / \mathrm{s}$ C.O.L.M $\Rightarrow 0=\frac{M}{3}(V+10 \sqrt{2})$ $\Rightarrow V=-10 \sqrt{2}$ So, $V$ is along $135^{\circ}$ ...
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Question: Light in certain cases may be considered as a stream of particles called photons. Each photon has a linear momentum $\mathrm{h} /{ }^{\lambda}$ where $\mathrm{h}$ is the Planck's constant and $\lambda$ is the wavelength of the light. A beam of light of wavelength $\lambda$ is incident on a plane mirror at an angle of incidence ${ }^{\theta}$. Calculate the change in the linear momentum of a photon as the beam is reflected by the mirror. Solution: $p_{i}=\frac{h}{\lambda}(\cos \theta \h...
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Question: A ball of mass $50 \mathrm{~g}$ moving at a speed of $2.0 \mathrm{~m} / \mathrm{s}$ strikes a plane surface at an angle of incidence $45^{\circ}$. The ball is reflected by the plane at equal angle of reflection with the same speed. Calculate (a) the magnitude of the change in momentum of the ball. (b) thechange in the magnitude of the momentum of the ball. Solution: $V_{i}=+v \cos 45^{\circ} \hat{\imath}+v \sin 45^{\circ} \hat{\jmath}$ $V_{f}=-v \cos 45^{\circ} \hat{\imath}+v \sin 45^{...
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Question: A man of mass $M$ having a bag of mass $m$ slips from a roof of a tall building of height $\mathrm{H}$ and starts falling vertically. When at a height $h$ from the ground, he notices that the ground below him is pretty hard, but there is a pond horizontal from the distance $x$ from the line of fall. In order to save himself he throws the bag horizontally in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in water. ...
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