Advertisements
Advertisements
प्रश्न
Answer the following question.
Obtain its value for an elastic collision and a perfectly inelastic collision.
Advertisements
उत्तर
- 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
संबंधित प्रश्न
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.
In an elastic collision of two bodies, the momentum and energy of each body is conserved.
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.
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?
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.
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
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 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.
Arrive at an expression for elastic collision in one dimension and discuss various cases.
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 ____________.
In Rutherford experiment, for head-on collision of a-particles with a gold nucleus, the impact parameter is ______.
In inelastic collision, ____________.
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 ______.
During inelastic collision between two bodies, which of the following quantities always remain conserved?
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.
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 rod of mass M and length L is lying on a horizontal frictionless surface. A particle of mass 'm' travelling along the surface hits at one end of the rod with velocity 'u' in a direction perpendicular to the rod. The collision is completely elastic. After collision, particle comes to rest. The ratio of masses `(m/M)` is `1/x`. The value of 'x' will be ______.
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 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)
The dimension of mutual inductance is ______.
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?
