Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
Given that, for every bounce, 64% of the initial energy is converted to final energy.
a) Coefficient of restitution in case of inelastic collision is given by,
e = `- "v"_"s"/"u"_"a" = - "v"/"u"` ....(1)
∴ `"e"^2 = "v"^2/"u"^2`
∴ v2 = e2 × u2
∴ `1/2 "mv"^2 = "e"^2 xx 1/2 "mu"^2`
∴ (K.E.)f = e2 × `1/2` mu2
∴ `("K.E.")_"f"/("K.E.")_"i" = "e"^2`
∴ `64/100 = "e"^2`
∴ e = 0.8
b) From equation (1),
v = – eu
∴ After first bounce,
v1 = - eu
after second bounce,
v2 = - ev1 = - e(- eu) = e2u
and after third bounce,
v3 = - ev2 = - e(- e2u) = - e3u
But u = `sqrt(2"gh")`
∴ `"v"_3 = - "e"^3 xx sqrt(2"gh") = -(0.8)^3 xx sqrt(2 xx 10 xx 5)` .....(∵ h = 5 m given)
= - (0.8)3 × 10 = - 5.12 m/s
c) Impulse given by the ball during third bounce, is,
J = Δp = mv3 - mv2
= m × (- e3u - e2u)
= - m × e2u × (e + 1)
= - 100 × 10-3 × (0.8)2 × 10 × (0.8 + 1)
= - 1.152 Ns
d) Average force exerted in 250 ms,
F = `"J"/"t" = (- 1.152)/(250 xx 10^-3)`
`= (- 0.1152)/25 xx 10^3`
= – antilog{log (0.1152) – log (25)} × 103
= – antilog{`bar1` .0615 – 1.3979} × 103
= – antilog{`bar3` .6636} × 103
= - 4.609 × 10-3 × 103
= - 4.609 N
e) Average pressure for area
0.5 cm2 = 0.5 × 10-4 m2
P = `"F"/"A" = (4.608)/(0.5 xx 10^-4)`
= 9.216 × 104 N/m2
APPEARS IN
संबंधित प्रश्न
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.
Define coefficient of restitution.
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 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.
Explain the characteristics of elastic and inelastic collision.
Arrive at an expression for elastic collision in one dimension and discuss various cases.
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 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 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 ______.
During inelastic collision between two bodies, which of the following quantities always remain 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 as shown in figure.
If the collision is elastic, which of the following (Figure) is a possible result after collision?
Two blocks M1 and M2 having equal mass are free to move on a horizontal frictionless surface. M2 is attached to a massless spring as shown in figure. Iniially M2 is at rest and M1 is moving toward M2 with speed v and collides head-on with M2.
- While spring is fully compressed all the KE of M1 is stored as PE of spring.
- While spring is fully compressed the system momentum is not conserved, though final momentum is equal to initial momentum.
- If spring is massless, the final state of the M1 is state of rest.
- If the surface on which blocks are moving has friction, then collision cannot be elastic.
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 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 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?
A bag of sand of mass 9.8 kg is suspended by a rope. A bullet of 200 g travelling with speed 10 ms-1 gets embedded in it, then loss of kinetic energy will be ______.
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 ______.
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 ______.
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an inelastic collision of two balls ?
What do the objects do "after collision"?
Which of the following real-life scenarios is the best example of a collision as defined in the source?
