Advertisements
Advertisements
Question
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?
Advertisements
Solution
- A bullet of mass m1 travelling with a velocity u, striking a stationary wooden block of mass m2 and getting embedded into it is a case of perfectly inelastic collision.
- In a perfectly inelastic collision, although there is a loss in kinetic energy, the principle of conservation of energy is not violated as the total energy of the system is conserved.
Loss in the kinetic energy during a perfectly inelastic head-on collision:
- Let two bodies A and B of masses m1 and m2 moving with initial velocity `vec"u"_1 and vec"u"_2` respectively such that particle A collides headon with particle B i.e., `"u"_1 > "u"_2`.
- If the collision is perfectly inelastic, the particles stick together and move with a common velocity `vec"v"` after the collision along the same straight line.
loss in kinetic energy = total initial
kinetic energy – total final kinetic energy, - By the law of conservation of momentum, m1u1 + m2 u2 = (m1 + m2) v
∴ v = `("m"_1"u"_1 + "m"_2"u"_2)/("m"_1 + "m"_2)` - Loss of kinetic energy,
`Delta "K.E" = (1/2"m"_1"u"_1^2 + 1/2"m"_2"u"_2^2) - 1/2("m"_1 + "m"_2)"v"^2`
`= (1/2"m"_1"u"_1^2 + 1/2"m"_2"u"_2^2) -1/2("m"_1 + "m"_2)[("m"_1"u"_1 + "m"_2"u"_2)/("m"_1 + "m"_2)]^2`
`= ("m"_1^2"u"_1^2 + "m"_1"m"_2"u"_2^2 + "m"_1"m"_2"u"_1^2)/(2("m"_1 + "m"_2)) + ("m"_2^2 "u"_2^2 - "m"_1^2"u"_1^2 - "m"_2^2"u"_2^2 - 2"m"_1"m"_2"u"_1"u"_2)/(2("m"_1 + "m"_2))`
`= ("m"_1"m"_2)/(2("m"_1 + "m"_2)) ("u"_1 - "u"_2)^2` - Both the masses and the term `("u"_1 - "u"_2)^2` are positive. Hence, there is always a loss in a perfectly inelastic collision. For a perfectly inelastic collision, as e = 0, the loss is maximum.
APPEARS IN
RELATED QUESTIONS
In an inelastic collision of two bodies, the quantities which do not change after the collision are the ______ of the system of two bodies.
State if the following statement is true or false. Give a reason for your answer.
Total energy of a system is always conserved, no matter what internal and external forces on the body are present.
State if the following statement is true or false. Give a reason for your answer.
In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
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?
Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following figure is a possible result after collision?
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 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.
Which of the following potential energy curves in Fig. cannot possibly describe the elastic collision of two billiard balls? Here r is distance between centres of the balls.
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.
Solve the following problem.
A ball of mass 100 g dropped on the ground from 5 m bounces repeatedly. During every bounce, 64% of the potential energy is converted into kinetic energy. Calculate the following:
- Coefficient of restitution.
- The speed with which the ball comes up from the ground after the third bounce.
- The impulse was given by the ball to the ground during this bounce.
- Average force exerted by the ground if this impact lasts for 250 ms.
- The average pressure exerted by the ball on the ground during this impact if the contact area of the ball is 0.5 cm2.
Explain the characteristics of elastic and inelastic collision.
In Rutherford experiment, for head-on collision of a-particles with a gold nucleus, the impact parameter 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 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 ____________.
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 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 ______.
A bullet fired from gun with a velocity 30 m/s at an angle of 60° with horizontal direction. At the highest point of its path, the bullet explodes into two parts with masses in the ratio 1:3. The lighter mass comes to rest immediately. Then the speed of the heavier mass is
In an elastic collision of two billiard balls, which of the following quantities remain conserved during the short time of collision of the balls (i.e., when they are in contact).
- Kinetic energy.
- Total linear momentum?
Give reason for your answer in each case.
A ball of mass m, moving with a speed 2v0, collides inelastically (e > 0) with an identical ball at rest. Show that for head-on collision, both the balls move forward.
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 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 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 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 ______.
The dimension of mutual inductance is ______.
Three identical blocks A, B and C are placed on horizontal frictionless surface. The blocks A and C are at rest. But A is approaching towards B with a speed 10 m/s. The coefficient of restitution for all collision is 0.5. The speed of the block C just after the collision is ______.
Which of the following real-life scenarios is the best example of a collision as defined in the source?
