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
The rate of change of total momentum of a many-particle system is proportional to the ______ on the system.
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.
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.
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.
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 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.
Arrive at an expression for elastic collision in one dimension and discuss various cases.
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 ______.
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 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 smooth sphere of mass 'M' moving with velocity 'u' directly collides elastically with another sphere of mass 'm' at rest. After collision, their final velocities are V' and V respectively. The value of V is given by ______.
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 as shown in figure.
If the collision is elastic, which of the following (Figure) is a possible result after collision?
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 ______.
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.
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 drunkard walking in a narrow lane takes 5 steps forward and 3 steps backward, followed again by 5 steps forward and 3 steps backward, and so on. Each step is 1 m long and required 1 s to cover. How long the drunkard takes to fall in a pit 13 m away from the start?
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)
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)
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an inelastic collision of two balls ?
