A ballistic pendulum is used to measure the speed of bullets. It comprises a heavy block of wood of mass M suspended by two long cords. A bullet of mass m is fired into the block horizontally. The block, with the bullet embedded in it, swings upward (Fig. P10.70). The center of mass of the combination rises through a vertical distance h before coming to rest momentarily. In a particular experiment, a bullet of mass 40.0 g is fired into a wooden block of mass 10.0 kg. The block–bullet combination is observed to rise to a maximum height of 20.0 cm above the block’s initial height. a. What is the initial speed of the bullet? b. What is the fraction of initial kinetic energy lost after the bullet is embedded in the block?
FIGURE P10.70
(a)
The initial speed of the bullet.
Answer to Problem 70PQ
The initial speed of the bullet is
Explanation of Solution
Write the expression of the conservation of linear momentum before and after collision.
Here,
Rearrange above equation to get
According to conservation of mechanical energy, kinetic energy of the bullet-block system immediately after collision is equal to gravitational potential energy of the bullet-block system at maximum displacement.
Write the mathematical expression for conservation of energy.
Here,
Write the expression for
Write the expression for
Here,
Put equations (III) and (IV) in equation (II) and rearrange it to get
Substitute
Conclusion:
Substitute
Therefore, the initial speed of the bullet is
(b)
The fraction of initial kinetic energy lost after the bullet is embedded in the block.
Answer to Problem 70PQ
The initial kinetic energy of the bullet is lost by
Explanation of Solution
The collision of bullet with block results in loss of some initial kinetic energy so that final kinetic energy after impact might be less than initial kinetic energy.
Initial kinetic energy of the system is equal to kinetic energy of the bullet before collision.
Write the expression for the initial kinetic energy.
Lose of kinetic energy is equal to difference between the final kinetic energy after the impact and initial kinetic energy of the bullet.
Final kinetic energy after the impact is equal to final potential energy of the block-bullet system at maximum displacement position.
Write the expression for the final kinetic energy.
Substitute
Write the expression for the percentage change in kinetic energy.
Conclusion:
Substitute
Substitute
Substitute
Therefore, the initial kinetic energy of the bullet is lost by
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