Advertisements
Advertisements
Question
Answer the following question.
Obtain its value for an elastic collision and a perfectly inelastic collision.
Advertisements
Solution
- Consider a head-on collision of two bodies of masses m1 and m2 with respective initial velocities u1 and u2. As the collision is head-on, the colliding masses are along the same line before and after the collision. The relative velocity of approach is given as,
ua = u2 - u1
Let v1 and v2 be their respective velocities after the collision. The relative velocity of recede (or separation) is then vs = v2 – v1
∴ e = `- "v"_"s"/"u"_"a" = - ("v"_2 - "v"_1)/("u"_2 - "u"_1) = ("v"_1 - "v"_2)/("u"_2 - "u"_1)` .....(1) - For a head-on elastic collision, According to the principle of conservation of linear momentum,
Total initial momentum = Total final momentum
∴ m1u1 + m2u2 = m1v1 + m2v2 ...(2)
∴ m1(u1 - v1) = m2(v2 - u2) ......(3)
As the collision is elastic, the total kinetic energy of the system is also conserved.
∴ `1/2 "m"_1"u"_1^2 + 1/2"m"_2"u"_2^2 = 1/2 "m"_1"v"_1^2 + 1/2 "m"_2"v"_2^2` .....(4)
∴ `"m"_1("u"_1^2 - "v"_1^2) = "m"_2("v"_2^2 - "u"_2^2)`
∴ m1(u1 + v1)(u1 - v1) = m2(v2 + u2)(v2 - u2) .....(5)
Dividing equation (5) by equation (3), we get
u1 + v1 = u2 + v2
∴ u2 - u1 = v1 - v2 .....(6)
Substituting this in equation (1),
e = `("v"_1 - "v"_2)/("u"_2 - "u"_1)` = 1 - For a perfectly inelastic collision, the colliding bodies move jointly after the collision, i.e.,
v1 = v2
∴ v1 - v2 = 0
Substituting this in equation (1),
e = 0
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:
Is the total linear momentum conserved during the short time of an elastic collision of two balls?
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 bullet of mass 0.012 kg and horizontal speed 70 m s–1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.
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.
Answer the following question.
A bullet of mass m1 travelling with a velocity u strikes a stationary wooden block of mass m2 and gets embedded into it. Determine the expression for loss in the kinetic energy of the system. Is this violating the principle of conservation of energy? If not, how can you account for this loss?
Solve the following problem.
A marble of mass 2m travelling at 6 cm/s is directly followed by another marble of mass m with double speed. After a collision, the heavier one travels with the average initial speed of the two. Calculate the coefficient of restitution.
Explain the characteristics of elastic and inelastic collision.
Define the following:
Coefficient of restitution
Two different unknown masses A and B collide. A is initially at rest when B has a speed v. After collision B has a speed v/2 and moves at right angles to its original direction of motion. Find the direction in which A moves after the 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 moving with velocity 5 m/s collides head on with another stationary ball of double mass. If the coefficient of restitution is 0.8, then their velocities (in m/s) after collision will be ____________.
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 bomb of mass 9 kg explodes into two pieces of mass 3 kg and 6 kg. The velocity of mass 3 kg is 16 m/s, The kinetic energy of mass 6 kg is ____________.
In inelastic collision, ____________.
A particle of mass 'm' collides with another stationary particle of mass 'M'. A particle of mass 'm' stops just after collision. The coefficient of restitution is ______.
During inelastic collision between two bodies, which of the following quantities always remain conserved?
A cricket ball of mass 150 g moving with a speed of 126 km/h hits at the middle of the bat, held firmly at its position by the batsman. The ball moves straight back to the bowler after hitting the bat. Assuming that collision between ball and bat is completely elastic and the two remain in contact for 0.001s, the force that the batsman had to apply to hold the bat firmly at its place would be ______.
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.
Two pendulums with identical bobs and lengths are suspended from a common support such that in rest position the two bobs are in contact (Figure). One of the bobs is released after being displaced by 10° so that it collides elastically head-on with the other bob.
- Describe the motion of two bobs.
- Draw a graph showing variation in energy of either pendulum with time, for 0 ≤ t ≤ 2T, where T is the period of each pendulum.
A ball of mass 10 kg moving with a velocity of 10`sqrt3` ms–1 along the X-axis, hits another ball of mass 20 kg which is at rest. After collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along Y-axis at a speed of 10 m/s. The second piece starts moving at a speed of 20 m/s at an angle θ (degree) with respect to the X-axis.
The configuration of pieces after the collision is shown in the figure.
The value of θ to the nearest integer is ______.
A ball falls from a height of 1 m on a ground and it loses half its kinetic energy when it hits the ground. What would be the total distance covered by the ball after sufficiently long time?
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 ______.
What is a collision?
