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
Answer
Ans. (c)
Total initial kinetic energy = (1/2)m1v02 + (1/2)m2v02 + (1/3)m3v02=3mv02
maximum speed will be realize if other particle speed are zero
So (1/2)m1(v1max)2 = 3mv02
Substituting for all m1, m2, m3
we get
v1max = 2.45v0
v2max = 1.73v0
v3max = 1.41v0
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
Answer
Ans. (a)
Time in first fall= √(2h/g)
now velocity of ball when it strike the floor
mgh = (1/2)mv2
v = √(2gh)
velocity of rebound = e√(2gh)
So time for the body to come again to floor= 2e√(2gh)/g
So Time till now = √(2h/g) + 2e√(2h/g)
velocity for the third rebound= e2√(2gh)
Time = 2e2√(2gh)/g
So Time till now = √(2h/g) + 2e√(2h/g)+ 2e2√(2gh)/g
So Total time
=√(2h/g) + 2e√(2h/g)+ 2e2√(2gh)/g + ……… =√2hg(1+2∑∞1en) =√2hg(1+e1−e)
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 vi=(vcosθ)i−(vsinθ)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) vf=(evcosθ)i−(vsinθ)j
(b) vf=(vcosθ)i+(evsinθ)j
(c) vf=(vcosθ)i−(evsinθ)j
(d) vf=(evcosθ)i+(vsinθ)j
Answer
Answer is (b)
Now vfsinθf=evsinθ --(1) vfcosθf=vcosθ --(2)
So final velocity vector vf=(vcosθ)i+(evsinθ)j
Question 4.
Find the final angle made with the horizontal
(a) tan−1(etanθ)
(b) cot−1(etanθ)
(c) tan−1(esinθ)
(d) tan−1(ecosθ)
Answer
Answer is (a)
From equation (1) and (2) tanθf=evsinθvcosθ=etanθ
Or θf=tan−1(etanθ)
Question 5.
Find the change in momentum Δp due to the impact
(a)mv(1+e)(cosθ)j
(b) mv(1−e)(sinθ)j
(c) mv(1−e)(cosθ)j
(d) mv(1+e)(sinθ)j
Answer
(d) is the answer vi=(vcosθ)i−(vsinθ)j vf=(vcosθ)i+(evsinθ)j
Δp=mvf−mvi
Substituting the values Δp=mv(1+e)(sinθ)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−e2)sin2θ
Answer
(a) ,(c) ,(d) are correct
Since frictional force are null, horizontal components of the momentum remain conserved.
The force acts in upward direction, as change in momentum is in upward direction
FΔt=Δp F=ΔpΔt
As Δp -> j direction ΔF -> j direction
Question 7.
Let p1 and p2 be the momentum of two equal particles before elastic collision and p’1 and p2’ be the momentum after collision.
Mow we know p2 = 0
which of the following is true
(a) p1.p1=p′1.p′1+p′2.p′2
(b) p1=p′1+p′2
(c) p′1.p′2=0
(d) none of the above
Answer
(a), (b), (c)
(a) is the law of conservation of energy p1.p1=p′1.p′1+p′2.p′2
(b) is law of conservation of linear momentum
p1=p′1+p′2 ---(2)
Squaring equation (2) p21=p′21+p′22+2p′1.p′2
Or p′1.p′2=0
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
Answer
(c). Total linear momentum remains conserved
Question 9. In a collinear collision, a particle with an initial speed v0 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) √2v0
(b) v02
(c) v0√2
(d) v04
Answer
Answer is (a)
From conservation of momentum mv0=mv1+mv2
or v0=v1+v2 -(1)
Given from questions 12mv21+12mv22=3212mv20 v21+v22=32v20 -(2)
Squaring (1) v20=v21+v22+2v1v2
or 2v1v2=−v202
Now (v1−v2)2=v21+v22−2v1v2=2v20
or |v1−v2|=√2v0
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(12mv2) then f=mM+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
Answer
Maximum energy loss happens when particles stick together
Then
mv = (M+m)V
or V=mvM+m
Now
Loss of KE = 12(M+m)V2−12mv2
Substituting the values of V from above
Loss of KE=Mm+M12mv2
Hence (d) is correct option
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=(n−1)Lv
(b) The last block starts moving at t=n(n−1)L2v
(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 vn
Answer
Since collision is perfectly inelastic so all the blocks will stick together one by one and move in a form of combined mass.
Time required to cover a distance L by first block =L/v
Now first and second block will stick together and move with v/2 velocity (by applying conservation of momentum) and combined system will take time L/v/2=2L/v to reach up to block third.
Now these three blocks will move with velocity v/3 and combined system will take time L/v/3=3L/v to reach upto the block fourth.
So, total time =Lv+2Lv+3Lv+...+(n−1)Lv=n(n−1)L2v
Velocity of combined system having n blocks as v/n.
Hence (b) is the answer
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
Answer
(d) is the answer
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
Answer
Let v1 and v2 be the velocities after the collision
Law of conservation of linear momentum m1u=m1v1+m2v2 ---(1)
For elastic collision
Velocity of approach=-velocity of separation u=−(v1−v2)
or v2−v1=u ---(2)
Solving equation (1) and (2)
We get v1=(m1−m2m1+m2)u
v2=(2m1m1+m2)u
Initial Kinetic energy (K1i ) =12m1u2
KE of the mass m1( K1f) =12m1v21 =12m1(m1−m2m1+m2)2u2
KE of mass m2(K2f ) =12m2v22 =12m2(2m1m1+m2)2u2
Now K2fK1i=12m2(2m1m1+m2)2u212m1u2 =4m1m2(m1+m2)2 =1-\frac{(m_1-m_2)^2}{(m{_1^}+m_2)^2}
Now when m1=m2 K2f=K1i
Velocity of center of mass before collision =m1um1+m2
Since no external force is acting, Center of mass velocity will not change
Velocity of center of mass after collision =m1um1+m2
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
Answer
Now we know from projectile motion
Velocity of strike =(ucosα)i−(usinα)j
Since no force acts in the horizontal direction, Horizontal velocity will be same be same after
Vertical velocity will experience the force
So vertical velocity after rebound =eusinαj
So velocity of particle after first rebound =(ucosα)i+(eusinα)j
Now in case of Projectile motion
Time of flight =2usinαg=2uyg
So time before striking first T1=2usinαg
Time taken after first strike and before second strike T2=2eusinαg
Now velocity of particle after second strike is given by =(ucosα)i+(e2usinα)j
Time taken after second strike and before third strike T3=2e2usinαg
Similarly other can find out
So total time =T1+T2+T3+... =2usinαg(1+e+e2+................) =2usinαg(11−e)
Now range is given by R=uxT1+uxT2+.... =uxT =ucosα[2usinαg(11−e)] =u2sin2αg(1−e)
Velocity after mth rebound =(ucosα)i+(emusinα)j
Tangent of angle of projection after mth rebound =emusinαucosα =emtanα
We know from projectile Height=u2sin2α2g
So height after m rebound =(emusinα)22g
Impluse exerted by the surface on the ball I1=m[eusinαj(−usinαj)] =musinα(1+e)j
Similarly I2=musinα(e+e2)j
I3=musinα(e2+e3)j
So total impulse =musinα[1+2(e+e2+e3+.......)] =musinα(1+e1−e)
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