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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Chapter 10 Solutions
Physics for Scientists and Engineers: Foundations and Connections
- Two bumper cars at the county fair are sliding toward one another (Fig. P11.54). Initially, bumper car 1 is traveling to the east at 5.62 m/s, and bumper car 2 is traveling 60.0 south of west at 10.00 m/s. They collide and stick together, as the driver of one car reaches out and grabs hold of the other driver. The two bumper cars move off together after the collision, and friction is negligible between the cars and the ground. a. If the masses of bumper cars 1 and 2 are 596 kg and 625 kg respectively, what is the velocity of the bumper cars immediately after the collision? b. What is the kinetic energy lost in the collision? c. Compare your answers to part (b) from this and Problem 54. Is one answer larger than the other? Discuss and explain any differences you find.arrow_forwardThree runaway train cars are moving on a frictionless, horizontal track in a railroad yard as shown in Figure P11.73. The first car, with mass m1 = 1.50 103 kg, is moving to the right with speed v1 = 10.0 m /s; the second car, with mass m2 = 2.50 103 kg, is moving to the left with speed v2 = 5.00 m/s, and the third car, with mass m3 = 1.20 103 kg, is moving to the left with speed v3 = 8.00 m /s. The three railroad cars collide at the same instant and couple, forming a train of three cars. a. What is the final velocity of the train cars immediately after the collision? b. Would the answer to part (a) change if the three cars did not collide at the same instant? Explain. FIGURE P11.73arrow_forwardInitially, ball 1 rests on an incline of height h, and ball 2 rests on an incline of height h/2 as shown in Figure P11.40. They are released from rest simultaneously and collide elastically in the trough of the track. If m2 = 4 m1, m1 = 0.045 kg, and h = 0.65 m, what is the velocity of each ball after the collision?arrow_forward
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