Elastic and Inelastic Collisions problems and solutions

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Multiple Choice Questions

Question 1.
Three bodies form an isolated system. There are m1 = m2 = 2m and m3 = 3m. They have different direction, but all have the same initial speed v0. One or more elastic collision between the pair of the bodies where otherwise do not intersect. Find the maximum possible final speed of each of the three bodies.
(a) 3v0, 2v0, v0
(b) v0, 2v0, 3v0
(c) 2.4v0, 1.73v0, 1.41v0
(d) none of the above

Question 2.
A ball is dropped from height h on a floor where Coefficient of restitution is e. Find the time required by the ball to stop rebounding
(a) √(2h/g) (1 + e/1 - e)
(b) √(2h/g) (1 + e)
(c) √(2h/g) (1 - e/1 + e)
(d) none of the above

Linked Type Comprehension

An object of mass M collides with a frictionless surface. The surface is assumed to be x-axis and object is coming at an angle θ to the x-axis. The object bounces from the surface but the collision is not elastic
The initial velocity vector of the object
$v_i=(vcos{\theta})\mathbf{i}-(vsin{\theta})\mathbf{j}$ The vertical component of the velocity undergoes a change due to the impact. The magnitude is fraction e of the original vertical components
The situation is depicted below in the figure

Question 3.
Find the velocity vector after the collision
(a) $v_f=(evcos{\theta})\mathbf{i}-(vsin{\theta})\mathbf{j}$
(b) $v_f=(vcos{\theta})\mathbf{i}+(evsin{\theta})\mathbf{j}$
(c) $v_f=(vcos{\theta})\mathbf{i}-(evsin{\theta})\mathbf{j}$
(d) $v_f=(evcos{\theta})\mathbf{i}+(vsin{\theta})\mathbf{j}$

Question 4.
Find the final angle made with the horizontal
(a) $tan^{-1} (etan{\theta})$
(b) $cot^{-1}(etan{\theta})$
(c) $tan^{-1}(esin{\theta})$
(d) $tan^{-1}( ecos{\theta})$

Question 5.
Find the change in momentum $\Delta \mathbf{p}$ due to the impact
(a)$ mv(1+e)(cos{\theta})\mathbf{j}$
(b) $mv(1-e)(sin{\theta})\mathbf{j}$
(c) $mv(1-e)(cos{\theta})\mathbf{j}$
(d) $mv(1+e)(sin{\theta})\mathbf{j}$

Question 6.
Which one of the following is true about the case?
(a) Momentum remain conserved in the X –direction
(b) There is no loss of kinetic energy of the object
(c) The force during the collision acts in upward direction on the object
(d) The ratio of the final KE to the initial KE is $1-(1-e^2){sin}^2{\theta}$

Multiple Choice Questions

Question 7.
Let p1 and p2 be the momentum of two equal particles before elastic collision and p1 and p2 be the momentum after collision.
Mow we know p2 = 0
which of the following is true
(a) $\mathbf{p_1}.\mathbf{p_1}=\mathbf{p_1^\prime}.\mathbf{p_1^\prime}+\mathbf{p_2^\prime}.\mathbf{p_2^\prime}$
(b) $\mathbf{p_1}=\mathbf{p_1^\prime}+\mathbf{p_2^\prime}$
(c) $\mathbf{p_1^\prime}.\mathbf{p_2^\prime}=0$
(d) none of the above

Question 8.
During inelastic collision between two bodies, which of the following quantities always remain conserved?
(a) Total kinetic energy.
(b) Total mechanical energy.
(c) Total linear momentum.
(d) Speed of each body

Question 9.
In a collinear collision, a particle with an initial speed $v_0$ strikes a stationary particle of the same mass. If the final total kinetic energy is 50% greater than the original kinetic energy, the magnitude of the relative velocity between the two particles, after collision, is
(a) $\sqrt {2} v_0$
(b) $\frac {v_0}{2}$
(c) $\frac {v_0}{\sqrt {2}}$
(d) $\frac {v_0}{4}$

Question 10.
Statement - I : A point particle of mass m moving with speed v collides with stationary point particle of mass M. If the maximum energy loss possible is given as $f(\frac {1}{2} mv^2)$ then $f=\frac {m}{M+m}$
Statement 0 II : Maximum energy loss occurs when the particles get stuck together as a result of the collision.

(a) Statement-I is true, Statement-II is true, Statement-II is a correct explanation of Statement-I.
(b) Statement-I is true, Statement-II is true, Statement-II is a not correct explanation ofStatement-I.
(c) Statement-I is true, Statement-II is false.
(d) Statement-I is false, Statement-II is true

Question 11.
A set of n identical cubical blocks lies at rest parallel to each other along a line on a smooth horizontal surface. The separation between the near surfaces of any two adjacent blocks is L. The block at one end is given a speed v towards the next one at time t=0. All collisions are completely inelastic, then
(a) The last block starts moving at $t=\frac {(n-1)L}{v}$
(b) The last block starts moving at $t=\frac {n(n-1)L}{2v}$
(c) The centre of mass of the system will have a final speed v
(d) The centre of mass of the system will have a final speed $\frac {v}{n}$

Question 12.
A body of mass m moving with velocity v collides head on with another body of mass 2m which is initially at rest. The ratio of K.E. of colliding body before and after collision will be
(a) 1 : 1
(b) 2 : 1
(c) 4 : 1
(d) 9 : 1

Subjective Numericals

Question 13
A body A having mass m1 travelling  with velocity  u  makes an head on elastic collision with the stationary body  B of mass m2
(a) Calculate the final velocities of A and B
(b) Calculate the Final Kinetic energy of both the mass
(c) Calculate the ratio of the kinetic energy transferred to m2 to the original kinetic energy
(d) For what value of m2, all the energy is transferred to the stationary object
(e) Calculate the velocity of the Center of mass before collision and after collision

Question 14.
An imperfectly elastic particle is projected from a point in horizontal plane with velocity u at any angle α to the horizon. If e is the Coefficient of restitution
Let i and j are the unit vector across the x and y axis respectively
(a) Find the velocity of particle after first rebound
(b) Find the total time taken by the particle before stopping rebounding
(c) Find the total range
(d) Find the velocity at the mth rebound
(e)Find the tangent of  angle of projection at mth rebound
(g) Find the height reached after mth rebound
(f) Find the total impulse exerted by the surface on the ball

Question 15.
A body of mass 3 kg moving with a velocity (i +2j +3k) m/s collides with another body of mass 4 kg moving with a velocity (2i+j+k) in m/s.They stick together  .Find the velocity of the composite body.Find the Kinetic energy before collision and after collision

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