Tag: work, energy and power

Questions Related to work, energy and power

A ball of mass m moving with velocity V makes a head on  elastic collision with a ball of the same mass moving with velocity 2V towards it. Taking direction of V as positive velocities of the two balls after collision are

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

  1. -V and 2V

  2. 2V and -V

  3. V and -2V

  4. -2V and V

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Correct Option: B

Two billiard balls undergo a head-on collision. Ball 1 is twice as heavy as ball 2. Initially, ball 1 moves with a speed $v$ toward ball $2$ which is at rest. Immediately after collision, ball 1 travels at a speed of $v/3$ in the same direction. What type of collision has occured?

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

  1. inelastic

  2. elastic

  3. completely inelastic

  4. cannot be determined from the information given

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Correct Option: B
Explanation:

Solving the equation of conservation of momentum give us that the relative velocity of approach is equal to the relative velocity of separation. Hence coefficient of restitution is 1.
Which means collision is elastic. 

The collision of two balls of equal mass takes place at the origin of coordinates. Before collision, the components of velocities are $(V _x = 50 c m s^{-1},  V _{y} = 0)$ and $(V _{x} = -40 c m s^{-1}$ and $V _{y} = 30 c m s^{-1})$. The first ball comes to rest after collision. The velocity (components $V _{x}$ and $V _{y}$ respectively) of the second ball are

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

  1. 10 and 30 $c m s^{-1}$

  2. 30 and 10 $c m s^{-1}$

  3. 5 and 15 $c m s^{-1}$

  4. 15 and $5 c m s^{-1}$

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Correct Option: C

Which of the following does not hold when two particles of masses $m _1$ and $m _2$ undergo elastic collision?

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

  1. When $m _1 =m _2$ and $m _2$ is stationary, there is maximum transfer of kinetic energy in head an collision

  2. When $m _1=m _2$ and $m _2$ is stationary , there is maximum transfer of momentum in head on collision

  3. when $m _1 >> m _2$ and $m _2$ is stationary, after head on collision $m _2$ moves with twice the velocity of $m _1$

  4. When the collision is oblique and $m _1=m _2$ with $m _2$ stationary, after the collision the particle move in opposite directions.

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Correct Option: A

Two identical balls A and B collide head on elastically. If velocities of A and B, before the collision are +0.5 m/s and -0.3 m/s respectively, then their velocities, after the collision, are respectively

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

  1. -0.5 m/s and +0.3 m/s

  2. +0.5 m/s and +0.3 m/s

  3. +0.3 m/s and -0.5 m/s

  4. -0.3 m/s and +0.5 m/s

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Correct Option: C
Explanation:

When identical balls collide $elastically $ they just $exchange$ their $SPEEDS$ and get reversed.

This can be verified by applying the $momentum$ conservation and $energy$ conservation, yes energy remains
 conserved for elastic collisions.
so the exchanged velocities will be following $v _1=0.3m/s$ and $v _2=-0.5m/s$

For head-on collision between two colliding balls of equal radii $r$, the impact parameter is equal to

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

  1. $2r$

  2. $Zero$

  3. $More \ than \ 2r$

  4. $Less\ than \ 2r$

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Correct Option: A

In a one-dimensional collision between two particles, their relative velocity is $\bar{v _1}$ before the collision and $\bar{v _2}$ and the collision.

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

  1. $\bar{v _1} = \bar{v _2}$ if the collision is elastic.

  2. $\bar{v _1} = - \bar{v _2}$ if the collision is elastic.

  3. $|\bar{v _2}| = |\bar{v _1}|$ in all cases.

  4. $\bar{v _1} = - k \bar{v _2}$ in all cases, where k $\geq$ 1.

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Correct Option: B,C,D
Explanation:

If $ { v } _{ 1 }$ is relative velocity before collision.
if ${ v } _{ 2 } $ is relative velocity before collision.
$e\le 1\$
$ e=\dfrac { { v } _{ 1 } }{ { -v } _{ 2 } } $


so $\left| { v } _{ 1 } \right| \ge \left| { v } _{ 2 } \right| $
also due to impact the ratios of velocity get changed,relative velocities D is correct. also, for elastic collision e$=$1.
So, option B is correct only if both particles have equal masses,not in general.

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 impulse $J$ to $A$.

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

  1. The total momentum of the '$A\ plus\ B$' system is $p$ before and after the impact, and $(p - J)$ during the impact.

  2. During the impact, $A$ gives impulse $J$ to $B$.

  3. The coefficient of restitution is $\displaystyle \dfrac{2 J}{p} - 1$

  4. The coefficient of restitution is $\displaystyle \dfrac{ J}{p} + 1$

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Correct Option: B,C
Explanation:

Let  $u=$ speed of A before impact. Thus,  $p=mu$.
Let $v _1, v _2 = $ speeds of  $A$ and $B$ after impact.
$u = v _1 + v _2 $ and $v _1 - v _2 = - eu$
$u = v _1 + v _2$ and $v _1 - v _2 = - eu$


$\therefore v _1 = \dfrac{1}{2} u (1-e)$ and $v _2 = \dfrac{1}{2} u (1 + e)$

$J = mv _2 = m \displaystyle \left [ \dfrac{1}{2} u (1 + e) \right ] = \dfrac{1}{2} p (1 + e)$

$\Rightarrow e=\dfrac{2J}{p}-1$

A sphere of mass m moving with a constant velocity collides with another stationary sphere of same mass. The ratio of velocities of two spheres after collision will be, if the co-efficient of restitution is e:

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

  1. $\displaystyle \frac{1 - e}{1 + e}$

  2. $\displaystyle \frac{e - 1}{e + 1}$

  3. $\displaystyle \frac{1 + e}{1 - e}$

  4. $\displaystyle \frac{e + 1}{e - 1}$

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Correct Option: A
Explanation:

The law of conservation  of linear momentum tells us that the overall momentum before the collision must be equal to the overall momentum after a collision.

Since the spheres have identical masses, we can write

$mu + m\times 0 = mv _A + mv _B$

$u = v _A+v _B$

From the definition of the coefficient of restitution, we know that

$e = \dfrac{v _B - v _A}{u}$

solving above two equations

$e \times ( v _A+v _B) = v _B - v _A$

$v _B(1-e) = v _A (1+e)$

$\dfrac{v _A}{v _B} = \dfrac{1-e}{1+e}$

In head on elastic collision of two bodies of equal masses:

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

  1. the velocities are interchanged

  2. the speeds are interchanged

  3. the momentum are interchanged

  4. the faster body slows down and the slower body speeds up

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Correct Option: A,B,C,D
Explanation:

For a head-on collision with a stationary object of equal mass, the projectile will come to rest and the target will move off with equal velocity. Hence, the velocities are interchanged i.e. the speeds are interchanged which in turn interchanges the momentum. Also, if target have some velocity then the faster body slows down and the slower body speed up.