Advertisements
Advertisements
प्रश्न
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
उत्तर
- 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
संबंधित प्रश्न
The rate of change of total momentum of a many-particle system is proportional to the ______ on the system.
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?
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?
Answer the following question.
Obtain its value for an elastic collision and a perfectly inelastic collision.
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.
Solve the following problem.
A spring ball of mass 0.5 kg is dropped from some height. On falling freely for 10 s, it explodes into two fragments of mass ratio 1:2. The lighter fragment continues to travel downwards with a speed of 60 m/s. Calculate the kinetic energy supplied during the explosion.
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.
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 block of mass 'm' moving on a frictionless surface at speed 'v' collides elastically with a block of same mass, initially at rest. Now the first block moves at an angle 'θ' with its initial direction and has speed 'v1'. The speed of the second block after collision is ______.
In inelastic collision, ____________.
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
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.
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 :
An insect moves with a constant velocity v from one corner of a room to other corner which is opposite of the first corner along the largest diagonal of room. If the insect can not fly and dimensions of room is a × a × a, then the minimum time in which the insect can move is `"a"/"v"`. times the square root of a number n, then n is equal to ______.
A sphere of mass 'm' moving with velocity 'v' collides head-on another sphere of same mass which is at rest. The ratio of final velocity of second sphere to the initial velocity of the first sphere is ______. ( e is coefficient of restitution and collision is inelastic)
What do the objects do "after collision"?
Before collision, what is the position of objects?
Which of the following real-life scenarios is the best example of a collision as defined in the source?
