Advertisements
Advertisements
Question
Answer the following question.
Discuss the following as special cases of elastic collisions and obtain their exact or approximate final velocities in terms of their initial velocities.
- Colliding bodies are identical.
- A very heavy object collides on a lighter object, initially at rest.
- A very light object collides on a comparatively much massive object, initially at rest.
Advertisements
Solution
The final velocities after a head-on elastic collision is given as,
`"v"_1 = "u"_1 [("m"_1 - "m"_2)/("m"_1 + "m"_2)] + "u"_2["2m"_2/("m"_1 + "m"_2)]`
`"v"_1 = "u"_1["2m"_1/("m"_1 + "m"_2)] + "u"_2[("m"_2 - "m"_1)/("m"_1 + "m"_2)]`
- Colliding bodies are identical
If m1 = m2, then v1 = u2 and v2 = u1. Thus, objects will exchange their velocities after head on elastic collision. - A very heavy object collides with a lighter object, initially at rest.
Let m1 be the mass of the heavier body and m2 be the mass of the lighter body i.e., m1 >> m2; the lighter particle is at rest i.e., u2 = 0 then,
m1 ± m2 ≅ m1 and `"m"_2/("m"_1 + "m"_2) ~= 0,`
∴ v1 ≅ u1 and v2 ≅ 2u1
i.e., the heavier colliding body is left unaffected and the lighter body which is struck travels with double the speed of the massive striking body. - A very light object collides on a comparatively much massive object, initially at rest.
If m1 is the mass of a light body and m2 is the mass of a heavy body i.e., m1 << m2 and u2 = 0. Thus, m1 can be neglected.
Hence v1 ≅ - u1 and v2 ≅ 0.
i.e., the tiny (lighter) object rebounds with the same speed while the massive object is unaffected.
APPEARS IN
RELATED QUESTIONS
State if the following statement is true or false. Give a reason for your answer.
In an elastic collision of two bodies, the momentum and energy of each body is conserved.
Answer carefully, with reason:
In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?
Answer carefully, with reason:
In an inelastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e., when they are in contact)?
Answer carefully, with reason:
If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy.)
A molecule in a gas container hits a horizontal wall with speed 200 m s–1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?
The bob A of a pendulum released from 30° to the vertical hits another bob B of the same mass at rest on a table, as shown in the figure. How high does the bob A rise after the collision? Neglect the size of the bobs and assume the collision to be elastic.
A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of 4 m s–1 relative to the trolley in a direction opposite to the its motion, and jumps out of the trolley. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run?
Consider the decay of a free neutron at rest : n → p + e–
Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in the β-decay of a neutron or a nucleus
Define coefficient of restitution.
Arrive at an expression for elastic collision in one dimension and discuss various cases.
What is inelastic collision? In which way it is different from an elastic collision. Mention a few examples in day-to-day life for inelastic collision.
A ball is thrown vertically down from height of 80 m from the ground with an initial velocity 'v'. The ball hits the ground, loses `1/6`th of its total mechanical energy, and rebounds back to the same height. If the acceleration due to gravity is 10 ms-2, the value of 'v' is
A ball of mass 0.1 kg makes an elastic head-on collision with a ball of unknown mass, initially at rest. If the 0 .1 kg ball rebounds at one-third of its original speed, the mass of the other ball is ______.
A mass M moving with velocity 'v' along x-axis collides and sticks to another mass 2M which is moving along Y-axis with velocity 3v. After collision, the velocity of the combination is ______.
A wooden block of mass 'M' moves with velocity 'v ' and collides with another block of mass '4M' which is at rest. After collision, the block of mass 'M' comes to rest. The coefficient of restitution will be ______.
A block of mass 'm' moving along a straight line with constant velocity `3vec"v"` collides with another block of same mass at rest. They stick together and move with common velocity. The common velocity is ______.
Two bodies of masses 3 kg and 2 kg collide bead-on. Their relative velocities before and after collision are 20 m/s and 5 m/s respectively. The loss of kinetic energy of the system is ______.
A body of mas 'm' moving with speed 3 m/s collides with a body of mass '2m' at rest. The coalesced mass will start to move with a speed of ______.
During inelastic collision between two bodies, which of the following quantities always remain conserved?
Consider a one-dimensional motion of a particle with total energy E. There are four regions A, B, C and D in which the relation between potential energy V, kinetic energy (K) and total energy E is as given below:
Region A : V > E
Region B : V < E
Region C : K > E
Region D : V > K
State with reason in each case whether a particle can be found in the given region or not.
The bob A of a pendulum released from horizontal to the vertical hits another bob B of the same mass at rest on a table as shown in figure.
If the length of the pendulum is 1 m, calculate
- the height to which bob A will rise after collision.
- the speed with which bob B starts moving. Neglect the size of the bobs and assume the collision to be elastic.
A particle of mass m with an initial velocity u`hat"i"` collides perfectly elastically with a mass 3m at rest. It moves with a velocity v`hat"j"` after collision, then, v is given by :
A ball is thrown upwards from the foot of a tower. The ball crosses the top of tower twice after an interval of 4 seconds and the ball reaches ground after 8 seconds, then the height of tower is ______ m. (g = 10 m/s2)
A bag of sand of mass 9.8 kg is suspended by a rope. A bullet of 200 g travelling with speed 10 ms-1 gets embedded in it, then loss of kinetic energy will be ______.
An alpha-particle of mass m suffers 1-dimensional elastic collision with a nucleus at rest of unknown mass. It is scattered directly backwards losing, 64% of its initial kinetic energy. The mass of the nucleus is ______.
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an inelastic collision of two balls ?
Which of the following real-life scenarios is the best example of a collision as defined in the source?
