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Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

A thin uniform rod of mass $m$ and length $l$ is hinged at the lower end of a level floor and stands vertically. It is now allowed to fall, then its upper and will strike the floor with a velocity given by(A)$\sqrt { mgl }$(B) $\sqrt { 3gl }$(c)$\sqrt { 5gl }$ (D) $\sqrt { 2gl }$  Sol. 

  1. A

  2. B

  3. C

  4. D
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using conservation of energy, the potential energy of the rod (mg*l/2) is converted into rotational kinetic energy (1/2 * I * omega^2). For a rod hinged at one end, I = ml^2/3. Thus, mg*l/2 = (1/2)*(ml^2/3)*omega^2. Solving for omega gives omega = sqrt(3g/l). The velocity of the tip is v = omega*l = sqrt(3gl).

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

Two balls of equal mass undergo head on collision while each was moving with speed $6\ m/s$. If the coefficient of restitution is $\dfrac{1}{3}$, the speed of each ball after impact will be

  1. $18\ m/s$

  2. $2\ m/s$

  3. $6\ m/s$

  4. $4\ m/s$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let speed of balls are $v _1$ and $v _2$.

There is no external force acting, momentum will be conserved. 
$m _1u _1+ m _2u _2= m _1v _1+m _2v _2$
$\Rightarrow m\times 6-m\times 6=mv _1+ mv _2$
$\Rightarrow v _1=-v _2$
Coefficient , $e=-\dfrac{v _1-v _2}{u _1-u _2}$  $\Rightarrow \dfrac{1}{3}= -\dfrac{v _1-v _2}{6+6}$  $\Rightarrow v _1=-2 m/s$

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

Two particles of masses ${m} _{1}$ and ${m} _{2}$ in projectile motion have velocities ${v} _{1}$ and ${v} _{2}$ respectively at time $t=0$. They collide at time ${t} _{0}$. Their velocities become ${v'} _{1}$ and ${v'} _{2}$ at time $2{t} _{0}$ while still moving in air. The value of $\left[ \left( { m } _{ 1 }{ v' } _{ 1 }+{ m } _{ 2 }{ v' } _{ 2 } \right) -\left( { m } _{ 1 }{ v } _{ 1 }+{ m } _{ 2 }{ v } _{ 2 } \right)  \right] $

  1. zero

  2. $({m} _{1}+{m} _{2})g{t} _{0}$

  3. $2({m} _{1}+{m} _{2})g{t} _{0}$

  4. $\cfrac{1}{2}({m} _{1}+{m} _{2})g{t} _{0}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

External force = $F _{ext}=\dfrac{\Delta p}{\Delta t}=((m _{1}v _{1}^{'}+m _{2}v _{2}^{'})-(m _{1}v _{1}+m _{2}v _{2}))/2t _{0}-0$

$((m _{1}v _{1}^{'}+m _{2}v _{2}^{'})-(m _{1}v _{1}+m _{2}v _{2})=2t _{0}F _{ext}=2to(m _{1}+m _{2})g$

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

A free hydrogen atom in ground state is at rest. A neutron of kinetic energy $K$ collides with the hydrogen atom. After collision hydrogen atom emits two photons in succession one of which has energy $2.55\ eV.$
(Assume that the hydrogen atom and neutron has same mass)

  1. Minimum value of $K$ is $25.5\ eV.$

  2. Minimum value of $K$ is $12.75\ eV.$

  3. The other photon has energy $10.2\ eV$ if $K$ is minimum.

  4. The upper energy level is of excitation energy $12.75\ eV$.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To excite the atom to the n=3 level (energy -1.51 eV from -13.6 eV), the energy required is 12.09 eV. If it emits a 2.55 eV photon (n=3 to n=2), the remaining energy must be 10.2 eV (n=2 to n=1). The minimum energy K must account for the excitation and the recoil energy of the atom.

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics
A cannonball is fired from a cannon at 50.0 m/s, at an angle of 30.0° above the horizontal on level ground. The goal is to determine the distance from its starting point to its ending point. The path of the cannonball is shown below. (inverse parabola).Calculate Δx.

  1. <span>15 cm</span>

  2. 20 cm&nbsp;

  3. 30 cm

  4. 10 cm
Reveal answer Fill a bubble to check yourself
D Correct answer
Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

In a one dimensional collision between two identical particles A and B, B is stationary and A has momentum P before impact. During impact B gives an impulse J to A. Then coefficient of restitution between the two is

  1. $\dfrac { 2J }{ P } +1$

  2. $\dfrac { 2J }{ P } -1$

  3. $\dfrac { J }{ P } +1$

  4. $\dfrac { J }{ P } -1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$p=mu$
$u=\dfrac{P}{m}$
$p-J$
$=V _A=\dfrac{P-J}{m}$     $V _B=\dfrac{J}{m}$

$\therefore \dfrac{v _{sep}}{V _{app}=\dfrac{V _B-V _A}{u}}$

$=\dfrac{\left(\dfrac{J}{m}\right)-\left(\dfrac{P-J}{m}\right)}{\dfrac{P}{m}}$

$=\dfrac{2J-P}{P}$

$=\dfrac{2J}{P}-1$
Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

A gun fires a shell of mass $1.5 $kg with velocity of $150$ m/s and recoils with a velocity of $2.5$ m/s. 

Calculate the mass of the gun.

  1. $20$ kg

  2. $30$ kg

  3. $90$ kg

  4. $60$ kg
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Mass of shell $=1.5$kg

Velocity of shell ${V _1} = 159\,m/s$
Recoil velocity $=2.5$ m/s
mass of gun $=M$
According to the conservation of momentum,
$m{v _1} = m{v _2}$
$\begin{array}{l} 1.5\times 150=M\times 2.5 \ M=\dfrac { { 1.5\times 159 } }{ { 2.5 } }  \ M=90\, kg \end{array}$

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

A particle of mass m, is attached to one end of a massless spring of force constant k, lying on a frictionless horizontal plane. The other end of the spring is fixed. The particle starts moving horizontally from its equilibrium position at time t = 0, with an initial velocity $u _0$. When the speed of the particle is $0.5\, u _0$, it collides elastically with a rigid walk. After this collision:

  1. The speed of the particle when it returns to its equilibrium position is $u _0$

  2. The time at which the particle passes through the equilibrium position for the first time is $t = \pi \sqrt{\dfrac{m}{k}}$

  3. The time at which the maximum compression of the spring occurs is $t = \dfrac{4\pi}{3}\sqrt{\dfrac{m}{k}}$

  4. The time at which the particle passes througout the equilibrium position for the second time is $t= \dfrac{4\pi}{3}\sqrt{\dfrac{m}{k}}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

In an elastic collision with a rigid wall, the particle reverses its velocity. Since the spring potential energy is conserved and the collision is elastic, the total energy remains constant. The particle will return to the equilibrium position with the same speed u0 it had initially.

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

A ball hits the floor and rebounds after an elastic collision. In this case:

  1. the momentum of the ball just after the collision is same as that just before the collision.

  2. the kinetic energy of the ball remains same during collision.

  3. the total momentum of the ball and the earth is conserved.

  4. the total energy of the ball and the earth remains the same.
Reveal answer Fill a bubble to check yourself
C,D Correct answer
Explanation

Since the velocity before and after the collision change hence momentum of the ball will change. So (a) is not true.  


Since the collision is inelastic a part of the mechanical energy is lost hence (b) is not true.  

Taking earth and the ball as a system there is no external force on the system. Hence the total momentum of the ball and the earth is conserved. So (c) is true.  

From the conservation principle of the energy, the total energy of the ball and the earth remains the same. Hence (d) is true.

Multiple choice collisions in one dimension collisions work, energy and power mechanics physics

Two identical spheres move in opposite direction with speed $v _1$ and $v _2$ and pass behind an opaque screen, where they may either cross without touching ( Event 1) or make an elastic head-on collision ( Event 2)

  1. We can never make out which events has occured

  2. We cannot make out which event has occured only if $v _1= v _2$

  3. We can always make out which event has occured

  4. We can make out which event has occured only if
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

For identical spheres colliding elastically, they exchange velocities. If they cross without touching, they continue with their original velocities. If they collide, they also end up with the same final velocities as if they had passed through each other. Thus, the final state is indistinguishable.