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
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 inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.
Answer carefully, with reason:
In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an elastic collision of two balls?
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.)
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.
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 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 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 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 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 particle of mass 'm' collides with another stationary particle of mass 'M'. A particle of mass 'm' stops just after collision. The coefficient of restitution 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 ______.
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 ______.
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 ______.
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.
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 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 ______.
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an inelastic collision of two balls ?
In a collision, what type of interaction occurs between objects?
What do the objects do "after collision"?
What is a collision?
Which of the following real-life scenarios is the best example of a collision as defined in the source?
