Advertisements
Advertisements
Question
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.
Advertisements
Solution 1
Initial speed of the bullet, ub = 70 m/s
Mass of the wooden block, M = 0.4 kg
Initial speed of the wooden block, uB = 0
Final speed of the system of the bullet and the block = ν
Applying the law of conservation of momentum:
mub + MuB = (m + M) v
0.012 × 70 + 0.4 × 0 = (0.012 + 0.4) v
∴ v = 0.84 / 0.412 = 2.04 m/s
For the system of the bullet and the wooden block:
Mass of the system, m' = 0.412 kg
Velocity of the system = 2.04 m/s
Height up to which the system rises = h
Applying the law of conservation of energy to this system:
Potential energy at the highest point = Kinetic energy at the lowest point
m'gh = (1/2)m'v2
∴ h = (1/2)(v2 / g)
= (1/2) × (2.04)2 / 9.8
= 0.2123 m
The wooden block will rise to a height of 0.2123 m.
Heat produced = Kinetic energy of the bullet – Kinetic energy of the system
= (1/2) mu2 - (1/2) m'v2 = (1/2) × 0.012 × (70)2 - (1/2) × 0.412 × (2.04)2
= 29.4 - 0.857 = 28.54 J
Solution 2
Here, m1 = 0.012 kg, u1 = 70 m/s
m2 = 0.4 kg, u2 = 0
As the bullet comes to rest with respect to the block, the two behave as one body. Let v be the velocity acquired by the combination.
Applying principle of conservation of linear momentum, (m1 + m2) v = m1H1 + m2u2 = m1u1
`v = (m_1u_1)/(m_1+m_2) = (0.012xx70)/(0.012+0.4) = (0.84)/0.412 = 2.04 ms^(-1)`
Let the block rise to a height h
P.E. of the combination = K.E. of the combination
`(m_1+m_2)gh = 1/2(m_1+m_2)v^2`
`h = v^2/(2g) = (2.04xx2.04)/(2xx9.8) = 0.212 m`
For calculating heat produced, we calculate energy lost (W), where
W = intial K.E. of bullet - final K.E of combination
`= 1/2m_1u_1^2 - 1/2(m_1+m_2)v^2`
`1/2xx0.012(70)^2 - 1/2(0.412)(2.04)`
W = 29.4 - 0.86 = 28.54 joule
:. Heat produced `H = W/J = (28.54)/4.2 = 6.8 cal`
RELATED QUESTIONS
The rate of change of total momentum of a many-particle system is proportional to the ______ on the system.
Answer carefully, with reason:
Is the total linear momentum conserved during the short time of an elastic collision of two balls?
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?
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.
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 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.
Define the following:
Coefficient of restitution
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 of mass 0.1 kg makes an elastic head-on collision with a ball of unknown mass, initially at rest. If the 0 .1 kg ball rebounds at one-third of its original speed, the mass of the other ball 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 ____________.
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
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 ______.
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 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)
